Chern numbers of algebraic varieties

Important news of developments in mathematics doesn't come out all that often, but here's some:

Chern numbers of algebraic varieties (6/10/09)

A problem at the interface of two mathematical areas, topology and algebraic geometry, that was formulated by Friedrich Hirzebruch, had resisted all attempts at a solution for more than 50 years. The problem concerns the relationship between different mathematical structures. Professor Dieter Kotschick, a mathematician at the Ludwig-Maximilians-Universität (LMU) in Munich, has now achieved a breakthrough. ... Kotschick has solved Hirzebruch's problem.

It would be somewhat hopeless to try to explain in a few paragraphs what this is all about. But since the result is a good example of the kinds of things mathematicians work on, a little discussion seems worthwhile.

A couple of important mathematicians of the 20th century are directly involved in the explanation. Friedrich Hirzebruch, who is still alive, worked in a number of active related fields, including topology, algebraic geometry, and complex manifolds. All of these are a legacy of the now well-known 19th century mathematician Bernhard Riemann.

Shiing-Shen Chern, who died in 2004 at the age of 93, worked primarily in another field closely related to those mentioned above, namely differential geometry.

The subject matter of all these fields comprises various types of abstract geometric objects. Perhaps the most familiar examples of such objects are plane curves defined by algebraic equations, such as circles, defined as the set of points (x,y) in the plane that satisfy x²+y²=R², where R is a constant (the radius of the circle).

Such curves can be generalized to any number of dimensions, as subsets in the space of n-tuples of numbers (x₁, ..., x_n) where all the coordinates x_i simultaneously satisfy some specific set of polynomial equations. The coordinates may be complex numbers (the set of which is denoted by ℂ). Such generalized curves in the space ℂⁿ are called algebraic varieties. The branch of mathematics that studies such things is called algebraic geometry.

Mathematicians like to classify things. In plane geometry, for example, polygons are classified as triangles, rectangles, pentagons, etc. according to the number of sides they have. All members of one of these classes of polygons are defined in terms of having the same number of distinct straight sides, say n.

A more general way in which geometric objects (or even more generally, topological objects) can be classified is in terms of objects which are all related by some sort of 1-to-1 mapping between their points. Many different types of mappings may be considered, such as mappings that preserve only the most basic topological properties (in the sense that donuts and coffee cups have the same "shape" in 3 dimensional space). Other types of mappings might be more narrowly defined, such as being "differentiable" (in the sense of calculus). A class of objects is then specified in terms of all objects that are related by some 1-to-1 mapping of a given type.

Normally, it is quite difficult to determine whether two objects are related by some mapping, and hence belong to the same class, because it's usually necessary to specify the mapping explicitly in order to do this.

But sometimes there are shortcuts that make the task of determining relatedness between topological objects easier. One of these is to find some number or set of numbers that is easily calculated and has to be the same for all members of the class of interest. For example, the class of n-gons in the plane is defined simply as all closed curves with the same number (n) of straight sides. Such numbers are called "invariants", because they are the same no matter what transformation (of the specified sort) is applied to any object of the class.

The best sort of invariant (or set of invariants) to have is one that is not only necessary for membership in the class, but also sufficient for membership. In that case one doesn't need to explicitly construct appropriate 1-to-1 mappings – one only has to calculate a few numbers (which may or may not actually be much easier).

Invariants need not be limited to ordinary numbers. They can also be more complicated sorts of mathematical objects, such as polynomials, groups, rings, or even topological objects like manifolds. All that one asks is that there be some way of calculating or determining the invariant associated with an object specified in some other way.

The "Chern numbers" mentioned above are important examples of such invariants, and the result which has now been proven is a statement about the properties of Chern numbers in a broad range of cases.

Why are mathematicians interested in such seemingly abstract constructs? Actually, there are many "real world" applications. Solutions of sets of polynomial equations – i. e., algebraic varieties – are often important in physics, especially mechanics. And more exotic sorts of topological objects crop up in problems involving differential equations.

Here's the research abstract:

Characteristic numbers of algebraic varieties

A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least 3, only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation preserving. In the space of Chern numbers, there are 2 distinguished subspaces, one spanned by the Euler and Pontryagin numbers, and the other spanned by the Hirzebruch–Todd numbers. Their intersection is the span of the Euler number and the signature.

The complete research paper can be found here.

Failure of unique factorization

For previous posts in this series, see here.

In this installment we're going to look at a detailed example and discuss some fine points of factorization of algebraic integers. This will be a bit long and pedantic. However, all the reasoning is elementary, except for a small amount of Galois theory, which you can review here.

Let's look at an example where unique factorization fails. First, we need to introduce a concept that makes it easy to prove some results about factorization (and has many more applications as well). Suppose we have an algebraic number α∈F and F⊇ℚ is a Galois extension. (Such an extension always exists: it is a splitting field of an irreducible polynomial f(x) such that f(α)=0, but we don't necessarily assume F is the smallest such extension.) Let G=G(F/ℚ) be the Galois group.

To review a concept, which has been introduced before, we define the norm of α with respect to the extension F⊇ℚ to be the product of all numbers σ(α) as σ ranges over elements of G. (More generally, the norm works also for Galois extensions of base fields that contain ℚ.) Symbolically, the norm is:

N_F/ℚ(α) = ∏_σ∈G σ(α)

Note that "the" norm depends on the specific extension F/ℚ, and so the extension is indicated in the subscript.

For instance, let F=ℚ(√-5). F/ℚ is Galois, because it is the splitting field of the irreducible polynomial f(x) = x²+5 = 0. Any α∈F can be written as a+b√-5 for a,b∈ℚ. An element σ∈G can be specified by how it acts on a typical such element. Of course, since [F:ℚ]=2, G has only two elements: 1 (the identity) and σ, so σ is determined by how it acts on √-5. σ(√-5) has to be a root of f(x)=0 different from √-5, so it must be -√-5. It follows that σ(a+b√-5)=a-b√-5 for a,b∈ℚ, because σ is a field automorphism of F that leaves all elements of ℚ fixed. This should remind you of complex conjugation, because that is in fact the nontrivial automorphism of the group G(ℚ(i)/ℚ).

In the simple case at hand, we can give a simple formula for the norm:

N_F/ℚ(a+b√-5) = (a+b√-5)(a-b√-5) = a² + 5b²

(In the field ℚ(i), the norm of a complex number is the square of the modulus, i. e. |a+bi|² = a² + b², so norm in the sense used here is closely related to the complex "norm".)

The norm symbol has some fairly obvious properties. From the way it is defined as a product, the norm is a group homomorphism from the multiplicative group of F to the multiplicative group of ℚ. It is also a homomorphism of the multiplicative semigroups of the rings of integers O_F and ℤ, which means that the value of the norm of an algebraic integer is always an integer in the base field (in this case the integers ℤ of ℚ). This is because each σ∈G is a field automorphism which satisfies σ(αβ)=σ(α)σ(β) for all α,β∈F. In other words,

N_F/ℚ(αβ) = N_F/ℚ(α) N_F/ℚ(β)

Furthermore, N_F/ℚ(ε)=±1 if and only if ε is an invertible element of O_F, i. e. a unit. (Since ±1 are the units of ℤ.)

Let's first determine the ring of integers O_F. Let α=a+b√-5 be a general element of F, with a,b∈ℚ. If in fact α is an algebraic integer, then so is its conjugate α*=a-b√-5. Further, the sum α+α*=2a is in ℚ, and is also an algebraic integer, since the algebraic integers of an extension form a ring. But the only algebraic integers in ℚ are in fact in ℤ, so 2a∈ℤ. Similarly, 2b=(α-α*)/√-5 is an algebraic integer in ℚ, hence an element of ℤ, so 2b∈ℤ. The norm of both α and α* is a²+5b², which is in ℤ. Multiplying the last expression by 4 shows (2a)²+(2b)²5∈4ℤ. Since 5≡1 (mod 4), (2a)²+(2b)²≡0 (mod 4). This is impossible unless both 2a and 2b are even integers (just check the separate cases). Hence both a and b are in ℤ. The conclusion is that O_F = {a+b√-5 | a,b∈ℤ}.

A slight elaboration of this argument shows that in any quadratic field ℚ(√d) where d∈ℤ is square-free, algebraic integers have the form O_F = {a+b√d | a,b∈ℤ} unless d≡1 (mod 4), in which case O_F = {(a+b√d)/2 | a,b∈ℤ, a≡b (mod 2)}. In other words, the naive guess that algebraic integers of ℚ(√d) just have the form a+b√d for a,b∈ℤ isn't entirely correct, but it is wrong only for d≡1 (mod 4), and then only by a little bit.

At this point, there is one delicate issue of nomenclature we must deal with. You will recall that a prime p∈ℤ is customarily defined as a (nonzero, nonunit) number which has no divisors other than units (±1) and ±p. We also proved that if p has this property, and if p divides a product mn, then either p divides m or p divides n (or maybe it divides both). In ℤ we can use this property to define p as a prime, since if the property is true of p then the more familiar condition that p has no nontrivial divisors is also true. This is because if p has this property, then the only divisors of p can be ±1 and ±p. (This follows from order properties of ℤ, because all divisors of a number n, except for ±n, have absolute values less than |n|.)

So these two properties of a nonzero p∈ℤ are equivalent. However, as we are about to see, the properties are not equivalent in other rings of integers. Nevertheless, we will find it convenient to use a generalization of the definition that p is prime if and only if p|mn implies p|m or p|m. So we will need a new term for the property of nonzero p that it is not a unit and not divisible by any other number except a unit times p. For this property we will use the term irreducible (like an irreducible polynomial). (And when this is the case, we say that p has only "trivial" factors, hence a nonunit p is defined to be irreducible if and only if all its factors are trivial, or equivalently if and only if it has no nontrivial factors.) We will make a similar distinction of "prime" and "irreducible" for integers α of other rings of integers.

Finally we can get back to factorization in F=ℚ(√-5). Observe that 21=3⋅7=(1+2√-5)(1-2√-5). We claim, first, that both 3 and 7 are irreducible in O_F. Consider 3 first. If α=a+b√-5 were a nontrivial integral divisor of 3 – i. e. neither α nor 3/α is a unit – then we would have N_F/ℚ(α) = a²+5b² divides N_F/ℚ(3) = 9. (Note, by the way, that for this extension, the norm is always nonnegative.) So N_F/ℚ(α) must be either 3 or 9, since α isn't a unit. Obviously the equation a²+5b²=3 has no solutions for a,b∈ℤ. So N_F/ℚ(α) isn't 3, hence it must be 9. Then N_F/ℚ(3/α)=1, and 3/α is a unit, contrary to assumption. So 3 is irreducible. 7 is also irreducible, by a similar argument. The same kind of argument shows that 1±2√-5 must be irreducible, since both conjugates have norm 21, and any non-unit α that divided either would have a norm equal to 3 or 7, which we just observed is impossible. And we cannot have 1±2√-5 dividing either 3 or 7 (or vice versa), since 21∤9 and 21∤49.

What we've just shown is that 21 has two factorizations into irreducible numbers of O_ℚ(√-5), and the factorizations are not equivalent, since the irreducible numbers in one factorization aren't unit multiples of either irreducible number in the other factorization. This shows that factorization of elements of the ring O_ℚ(√-5) into irreducible numbers isn't unique.

This example shows that a number which has no non-trivial factors (e. g. 3 or 7) can divide a product (e. g. 21) of two other numbers (e. g. 1±2√-5) without dividing either one of the factors of the product. So an irreducible number is not "prime" in the sense that if it divides a product, it must divide at least one of the factors. This latter property is actually more useful in practice, so we want to use the term "prime" for it. Therefore, a distinction is made in a general ring of integers: a (nonzero, nonunit) number which has no non-trivial factors is said to be irreducible. (Equivalently, if α=βγ then either β or γ must be a unit.) On the other hand, the term prime is reserved for (nonzero, nonunit) numbers α which have the property α|βγ implies α|β or α|γ.

Now, in any ring of integers of an algebraic number field, a prime integer (in the new sense) must also be irreducible. This is because if α is not irreducible, then by definition we can write α=βγ, where neither factor is a unit. But if α is prime it must divide one of its factors. Say α divides β. Then β=αδ. Hence 1=δγ. That is, γ is a unit, contrary to assumption, and so α has only trivial factors, so it's irreducible. Thus the set of prime integers is a subset of the set of irreducible integers.

However, in the example we just examined where F=ℚ(√-5), where we do not have unique factorization, then some irreducible numbers are not prime. E. g. 1+2√-5 is irreducible (as we showed), but it is not prime, because it divides 21, but does not divide either 3 or 7. Therefore it's possible for the set of prime integers to be a proper subset of the set of irreducible integers, i. e. a strictly smaller subset.

This raises some interesting questions. Recall that the cases we are interested in are the rings of algebraic numbers. By definition, these are the rings of integers A=O_F of a finite extension F/ℚ.

In any such A, it is always true that we can write any element as a product of a finite number of irreducible integers. The reason is that for any (nonzero, nonunit) α∈A, if α isn't irreducible, we can write α=βγ, where neither factor is 0 or a unit. Since F/ℚ is a finite extension, we can always compute norms, and we have N_F/ℚ(α) = N_F/ℚ(β)N_F/ℚ(γ). In some extensions a norm can be negative, but we can also stick in the absolute values of each term, and since no term is ±1, each factor on the right hand side is an an integer in ℤ that is strictly smaller in absolute value than |N_F/ℚ(α)|. Since all numbers here are finite, this process can't continue indefinitely. So not only do we get a finite product of irreducible integers, but we in fact get a finite product of finite powers of distinct irreducible integers. However, as the example above showed, this factorization need not be unique.

On the other hand, we can also say that if some algebraic integer α can be expressed as a product of powers of distinct prime integers, then (up to order and unit factors), the expression is unique as to which primes occur in the factorization and the powers of each that occur. To prove this, note first that any prime π which appears in one factorization into powers of primes must appear in the other. Because since π is in one factorization, it divides α, and because it is prime, it must divide at least one factor in any other factorization into powers of distinct primes. That factor must then be a power of π, since π can't divide a power of a different prime (using the fact that all primes are irreducible). Furthermore, if π occurs at all in some factorization of α, it must occur to eactly the same power in each factorization. Otherwise if the smaller power has exponent n, then we could cancel πⁿ from both factorizations. That would leave the integer α/πⁿ with distinct prime power factorizations, one containing π and the other not, which we just ruled out.

The problem here is that we do not know that every integer α∈A actually has even one factorization into distinct primes. Consequently, if there can be irreducible integers of A that aren't prime, so the set of prime integers is a proper subset of the set of irreducible integers, we cannot be sure that there is the kind of unique factorization theorem for integers of A that we have for ℤ, regardless of whether we specify "primes" or "irreducibles". Factorizations into irreducibles can't be guaranteed to be unique, while factorizations into only powers of primes might not even exist.

However, if the set of primes is the same as the set of irreducibles, then factorizations of any integer of A into irreducibles, and hence primes, are guaranteed to exist, And furthermore, the factorizations must also be unique.

What about converses? Suppose we can guarantee that factorizations of any α∈A into primes must exist. Does that imply prime = irreducible? Yes, for the following simple reason. We have already shown that if a factorization into primes exists, it must be unique. So suppose α is irreducible. If a factorization into primes must exist, then α=πβ for some prime π. But because α is irreducible, β has to be a unit. Any prime times a unit is still a prime, so α itself is a prime, and the set of irreducible elements contains only primes.

Or suppose we can guarantee that factorizations of any α∈A into irreducibles are unique. Does that imply prime = irreducible? Again the answer is yes. For suppose α is irreducible and that for some β and γ we have α|βγ. Since α|βγ there is a δ such that αδ = βγ. Write the right and left hand sides as a product of powers of distinct irreducible numbers, so that (ignoring possible factors which are units) αδ₁⋅⋅⋅δ_m = β₁⋅⋅⋅β_n (except that if α is among the δ_i, combine the terms). Then by the assumed uniqueness α^k = β_j for some j and some power k≥1, and both sides are powers of the same irreducible number α. That number must have been a divisor of either β or γ (or both). In any case, this means α is prime.

What we have now proven is this: If A=O_F is the ring of integers of a finite extension F/ℚ, the following conditions are equivalent:

1. The set of irreducible elements of A is the same as the set of prime elements of A (up to unit factors).
2. Every element of A has a unique factorization into powers of irreducible elements (up to unit factors).
3. Every element of A has a unique factorization into powers of prime elements (up to unit factors).

So (as is obvious) if all irreducible elements are prime, the difference in how these are defined is irrelevant. However, if there are irreducible elements that aren't prime, then factorizations of some integers into powers of irreducibles will not be unique, and some integers will not even have a factorization into powers of primes.

It turns out that there are certain types of rings in which all irreducible elements are prime, so the two concepts are equivalent in such rings. ℤ is one example of such a ring, but it is not the only one. Certainly, if we could guarantee that the integers of some extension of ℚ always had some factorization into primes (in the special sense used here), then as we showed, the factorizations must be unique. In order to investigate this issue further, we need help from the theory of ideals of rings of integers. By placing certain conditions on the types of ideals that the ring can have, we will be able to guarantee that any irreducible integer is prime, so that factorizations of any integers into irreducibles (which always exist) are also factorizations into primes, and therefore that they are unique.

One important type of ring that has this property is called a principal ideal domain, which means that every ideal consists of elements that are multiples of a single element (that isn't a unit) by some element of the ring. This is in fact the case with ℤ, where all ideals are of the form nℤ for some n. (The ideal is the full ring ℤ itself if n=±1.) But there are other rings of integers that are also principal ideal domaings, and a large part of algebraic number theory is about identifying which rings have this property. We'll look into this in much more detail, but first we need to explain further why we care about unique factorization.

Tags: algebraic number theory, unique factorization.

Algebraic number theory - index

This is a list of posts in the algebraic number theory series to date, oldest first.

1. A brief history of algebra

2. Numbers - rational and irrational, real and imaginary

3. Diophantine equations

4. Groups and rings

5. Fields and Galois theory

6. Modular arithmetic

7. Rings of algebraic integers

8. Rings and ideals

9. Uniqueness of factorization

10. Failure of unique factorization

11. Why do we care about unique factorization?

12. More concepts from ring theory

13. Factorization of prime ideals in extension fields

14. Splitting of prime ideals in quadratic extensions of ℚ, part 1

15. Splitting of prime ideals in algebraic number fields

16. Roots of unity and cyclotomic fields

17. Cyclotomic fields, part 2

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Tags: algebraic number theory

Uniqueness of factorization

Time for another installment of the series on algebraic number theory. Check here for previous articles.

In this installment we're going to look at an important property that some rings (such as ℤ) have, although most rings do not. But it is a useful and important property for proving many number theoretic results, which is why one bothers to consider it. We'll illustrate that soon.

But first we need a little terminology. In any ring, a unit is a ring element that has a multiplicative inverse which is also in the ring. For instance, in ℤ 1 and -1 are units, and they are the only units. Other rings of algebraic integers can have many units, and the set of units of the ring form an abelian group under multiplication. Determining this group of units, in fact, is one of the interesting computational issues in algebraic number theory.

Another important concept is that of a prime element of a ring. A little bit of care is required to define "prime" in a general ring, but essentially a prime element is one that has no factors other than itself and units. As far as divisibility and factors are concerned, units are essentially irrelevant, since they are invertible.

One of the most important properties that the integers have as a ring is unique factorization. That is, for any n∈ℤ, there is a unique way (apart from order and unit factors) to write n as a product of primes.

This fact can be proven using the order properties of ℤ, i. e. for every pair of distinct positive integers a, b, exactly one of a<b, a=b, or a>b is true. To begin with, this implies that for any pair of positive a,b∈ℤ, we can write a=qb+r with 0≤q and 0≤r<b. Reason: you can subtract b from a only a nonnegative but finite number of times (q) before the result is negative. This is because every number in the sequence a, a-b, a-2b, ... is strictly less than its predecessor, and if a is finite, there are only a finite number of distinct positive integers less than a. r is simply the last quantity before you have a negative number, and so 0≤r<b. The numbers q and r are uniquely determined by this procedure, and in fact there is a simple algorithm to find them, as we'll see in a moment.

For any positive integers a,b∈ℤ, we can define the greatest common divisor of the pair as the largest (positive) integer which divides both, written gcd(a,b), or simply (a,b). It may be, of course, that (a,b)=1, in which case we say a and b are relatively prime. As a matter of notation, if one number m divides another n, so that n=mq for some q∈ℤ, we write m|n. If this is not the case, then we write m∤n. (a,b) can be defined by the conditions that (a,b)|a, (a,b)|b, and if both c|a and c|b, then c|(a,b).

The Greek mathematician Euclid, known best for his geometry, was interested in number theory also. In addition to proving that there are infinitely many primes, he also gave a simple algrorithm for computing the greatest common divisior of two integers without explicitly factoring them – since factoring can be a relatively difficult process for large numbers. The algorithm is called, of course, the Euclidean algorithm.

To apply it, assume (without loss of generality) that a>b and write a=q₁b+r₁. Here, q₁>0 and 0≤r₁<b. Provided r₁≠0 we can repeat the procedure and write b=q₂r₁+r₂. We can repeat this procedure as long as the remainder r_k isn't 0. If r_k is the last nonzero remainder, then one notes that (a,b)|r_k, because in fact (a,b) divides all such remainders in the process. But we also have r_k-1=q_k+1r_k, hence r_k|r_k-1 and from r_k-2=q_kr_k-1+r_k, we find r_k|r_k-2 too. If we proceed back all the way we find r_k|b and r_k|a, hence r_k|(a,b). Therefore r_k=(a,b). In other words, (a,b) is the last nonzero remainder in this process.

But even nicer things are true. Go back to a=q₁b+r₁, so that r₁=a-q₁b. Similarly, r₂= b-q₂r₁= b-q₂(a-q₁b)= Ma+Nb for some integers M and N (not necessarily positive). Proceding inductively, we have that (a,b)=Ma+Nb for some M,N∈ℤ. What this says is that a certain Diophantine equation can be solved for unknowns M and N if a, b (and hence (a,b)) are given. Note that if (a,b)>1, the equation d=Ma+Nb could not be solved if 1≤d<(a,b), because a solution would imply (a,b)|d.

We need one more fact about prime numbers. Suppose p is prime, and p|mn for some m,n∈ℤ. So by definition, mn=pq for some q∈ℤ. We claim that p must divide either m or n (perhaps both). For suppose that we don't have p|m, hence (p,m) can't be p. But p is prime, and (p,m)|p, so we must have (p,m)=1. Hence it is possible to write 1=Mp+Nm. Therefore n=n(Mp+Nm)=Mnp+Nmn=Mnp+Npq= p(Mn+Nq). In other words, p|n. This property possessed by primes in ℤ is not shared by "primes" in other rings of algebraic integers, as we shall soon see.

We now have all the facts we need to prove unique factorization in ℤ. The proof is done by supposing factorization isn't unique, and showing this leads to a contradiction. So suppose factorization isn't unique, and for some n there are two different factorizations of n (apart from units ±1). There cannot be any prime which occurs in one factorization but not the other, by the result of the preceding paragraph. Hence the same prime factors occur, but for at least one prime p we have n=Ap^r=Bp^s with 0<r<s, and (A,p)=(B,p)=1. Dividing through by p^r reduces to the case where a prime occurs in one factorization but not the other, which is impossible. The contradiction proves the desired result.

It may seem "obvious" that factorization is unique, because we are so familiar with the fact this is true in ℤ that it is taken for granted. It may therefore be rather surprising that in many (in fact most) rings of algebraic integers, factorization is not unique. Unique factorization is actually a very special and rare occurrence, and a great deal of algebraic number theory is concerned with either trying to compensate for this "problem", or else trying to describe, in some sense, just how badly factorization fails to be unique.

In the next installment we'll explain why unique factorization is a useful property and look at some examples.

Tags: algebraic number theory, unique factorization.

Rings and ideals

It's time to do some more algebraic number theory again. For a refresher on what's gone on so far, check here.

In the last installment, way back in June, I introduced rings of algebraic integers, which are the main object of study in algebraic number theory. A "ring" in abstract algebra is a very fundamental concept, first discussed in this article. And in this article the most elementary example of a ring – the rational integers (ℤ) – was discussed, along with the concept of modular arithmetic.

What we saw there was the construction, for any n∈ℤ, of a new ring (ℤ/nℤ), which has only finitely many elements. Modular arithmetic is a staple of elementary number theory (that is, the classical theory of numbers, which deals mainly with ℤ). It was introduced by Carl Friedrich Gauss over 200 years ago, in 1801.

Now we are going to see how that construction can be generalized in abstract ring theory, using the concept of "ideals". It will turn out that the set nℤ consisting of all integer multiples of any n∈ℤ is an example of an ideal, and the "quotient ring" ℤ/nℤ can be generalized for any abstract ring and any ideal of that ring. This construction occurs ubiquitously in the study of rings of algebraic integers, introduced in the last installment.

An ideal can be defined for an arbitrary ring R, but it's a little messy. If R isn't commutative, there can be ideals which are "right" ideals but not "left" ideals or vice versa, because the definition of an ideal involves multiplication. So we'll assume R is commutative and has a multiplicative identity element ("1") too. For such a ring R, consider a subset I⊆R. Then I is an ideal just in case:

• I is closed under addition: a+b∈I for all a,b∈I.
  
• I is closed under multiplication by any element of R:
  ar=ra∈I for all a∈I and r∈R.
  

These axioms imply that I is a subgroup of R under addition. The additive identity 0 is in I by the second axiom. The second axiom also means that additive inverses are in I because R has a multiplicative identity, and hence its additive inverse -1∈R, so -a∈I for all a∈I. Note that if I≠R, I isn't a full ring (so it isn't a subring of R), because if 1∈I we would have I=R, by the second axiom.

One of the motivations for this concept of ideals is that it makes possible the definition of another very important concept: quotient rings. If as above R is a ring and I is an ideal, the quotient ring, denoted by R/I, is defined as the set of distinct cosets of the form r+I for all r∈R, where r+I is defined for any r∈ R as the set {r+a | a∈I}. Not all cosets are distinct as r ranges over R. Two cosets r+I and r′+I are the same exactly when r-r′∈I. This is because we can write r+I = r′+(r-r′)+I = r′+I. (Because r+I = I if r∈I.)

Importantly, we can define a ring structure on the set of all cosets. Addition is simple: (r+I)+(r′+I) = (r+r′)+I. Multiplication is a little trickier: (r+I)(r′+I) = rr′+I, but this only works since it doesn't depend on the choice of representative of each coset. The problem is we can have r+I=r′′+I even though r≠r′&prime, if r-r′′∈I. But in that case, (r-r′′)r′∈I by the second axiom for ideals, so all is OK. Thus multiplication of cosets is well-defined and unambiguous.

Under these definitions of addition and multiplication R/I is a (commutative) ring with a multiplicative identity. The additive identity element is I, and the multiplicative identity is the coset 1+I.

The ideal structure of the rational integers ℤ provides some important examples. Let n∈ℤ. Then obviously the set nℤ = {nm | m∈ℤ} is an ideal, often written simply as (n). It is just all integral multiples of n. The quotient ring ℤ/nℤ=ℤ/(n) is known as the ring of integers modulo n, and it has n elements. It is familiar from elementary number theory, where one writes "equations" such as a≡b (mod n) just in case a-b is divisible by n, i. e. is a multiple of n, i. e. a-b∈(n). The study of such congruences, which is done all the time in number theory, is really just the study of the ring ℤ/nℤ.

A very important special case is when n is a prime p. Then it is a fact that ℤ/pℤ is a field -- a finite field of p elements, sometimes denoted by F_p. However, if n is not prime, then ℤ/nℤ isn't even an integral domain, because it has divisors of zero, i. e. nonzero elements whose product is 0. For instance, if n=st for s,t∈ℤ, but s,t≠±1, then as ideals (s)≠0 and (t)≠0, where 0=(0)=(n). Yet (s)(t)=0. We can in fact characterize prime numbers p as elements of ℤ such that ℤ/pℤ is a field.

Tags: algebraic number theory, ring theory

Rings of algebraic integers

"The time has come," the Walrus said,
"To talk of many things:
Of shoes--and ships--and sealing-wax--
Of cabbages--and rings

Well, that's not exactly what he said, but close enough.

In our last installment of the series on algebraic number theory, we reviewed a couple of very important elementary examples of rings, namely the integers ℤ and the finite "quotient rings" ℤ/nℤ for any integer n>1. Earlier (here) we provided the basic abstract algebraic definition of a ring. In this and subsequent installments we're going to tackle rings in earnest, as they provide the best way to formalize the bulk of concepts needed to discuss algebraic number theory seriously.

Recall that our original objective was to find solutions to polynomial equations f(x)=0, and usually more specifically where the coefficients of the polynomial are ordinary integers in ℤ. Even more specifically, we were interested in Diophantine equations, where we wanted solutions that are integers, or "nearly" so.

Abstractly, we know by the fundamental theorem of algebra that if n is the degree of f(x), then exactly (counting multiplicity) n solutions exist in the complex numbers ℂ. If the coefficients of f(x) were rational numbers (in ℚ), then these solutions are by definition algebraic numbers. We know, further, that in some sense Galois theory provides a good description (so far as one can be given) of the nature of solutions of one polynomial equation in one variable with rational coefficients.

What we might hope for is finding solutions of such equations in integer or rational numbers. But that's very hard, so pragmatically it is best to ease off a bit and look for solutions in a somewhat larger class that's easier to handle theoretically. The key to the whole enterprise is finding the right abstraction to work with. We need to seek solutions which are conceptually somewhere between integers and fully general algebraic numbers. If we can deal with some intermediate construct, hopefully we can (with care) manage to bridge the gap and cross the chasm.

So let's call the concept we are looking for algebraic integers, and try to figure out how this concept should be defined. We have an advantage over mathematicians of the 19^th century who first thought about these issues. Namely, and as a direct result of their work, we now have a collection of abstract algebraic concepts that have proven very useful for organizing the theory. High in importance among these concepts is that of a ring. Since ℤ is a ring, if algebraic integers are to form a construct that generalizes ℤ, this construct had better be a ring if we dare call it a generalization.

The next point to consider is that what is to be treated as an algebraic integer depends very much on being relative to a specific finite extension F⊇ℚ. More explicitly, if F is a field which is a finite extension of ℚ (that is, has finite degree over ℚ), F is called a number field or algebraic number field, because all of its members are algebraic numbers as defined previously (roots of some polynomial f(x)∈ℚ[x]).

To give a name to that which we are looking to define, let O_F (or O_F/ℚ when we want to be explicit about the base field) be the set of all elements of F that are algebraic integers. In short, F is a field of algebraic numbers over ℚ, and O_F is to be a ring that plays a role in F analogous to the role of ℤ in ℚ. So we need to have ℤ⊆ O_F.

For a clue as to how to define O_F, suppose F=ℚ. F could then be defined as the set of solutions of f(x)=0, where f(x) is a first degree polynomial with coefficients in ℤ. That is, f(x) has the form ax+b, for a,b∈ℤ and a≠0. In other words, F is just all fractions b/a with a,b∈ℤ and a≠0, namely ℚ.

In this trivial case, how are the integers defined? All we need to do is require that f(x) be a monic first degree polynomial with coefficients in ℤ, that is f(x) has the form x+a for a∈ℤ. This suggests defining O_F as the set of elements of F that have a minimal polynomial f(x) which is monic and has coefficients in ℤ. Somewhat amazingly, this turns out to work nicely.

With this definition, ℤ⊆O_F obviously. What isn't so obvious and needs to be shown is that O_F is a ring. In particular, if a,b∈O_F, then both a+b and ab are also in O_F. That's a little work, but not hard. It can even be shown that F is the "field of fractions" of O_F, namely the set of all quotients b/a with a,b∈O_F and a≠0. Beyond that, there are a whole slew of other results that flow from this definition and justify it many times over.

So rings of integers as defined here are the natural generalization of the rational integers ℤ, which is a subring of any ring of algebraic integers. It is fair to say that the main concern of algebraic number theory is determining properties of such rings O_F for algebraic number fields F. However, there are various properties ℤ has which general rings of integers do not have. For example, all ideals of ℤ are "principal" ideals, and all elements of ℤ factor uniquely as a product of primes. Most rings of algebraic integers have neither of these properties, but criteria can be given for when they are present.

For future reference, note that we can also talk about rings of integers of arbitrary algebraic extensions E⊇F, where the base field isn't necessarily ℚ. The definition of the integers of E/F, denoted by O_E/F, is simply all elements of E having a minimal polynomial over F which is monic and has coefficients in O_F. This is useful for the general theory, but harder to work with when developing the basic theory.

To conclude this installment, let's look at the simplest sort of extension of ℚ, a quadratic extension F=ℚ(√d), where d is a square-free integer, which may be either positive or negative. Now √d is an algebraic integer because it satisfies x²-d=0. So is, for any b∈ℤ, b√d, since it satisfies x²-db²=0.

However, the situation for a+b√d, a,b∈ℤ isn't quite as obvious, unless we use the general fact (that hasn't been proven here) that the algebraic integers of F⊇ℚ form a ring, so that sums of integers are integers. It would be clearer just to produce the equation a+b√d satisfies in order to see directly that it is an integer. Fortunately this is quite easy.

Let α=a+b√d. From Galois theory, we recall that the "conjugate" of α is α*=a-b√d, and we have (x-α)(x-α*)=0. Multiplying things out we have f(α)=0, where f(x)=x²-2ax+a²-db². Clearly f(x) is monic with integer coefficients, so a+b√d is an algebraic integer in ℚ(√d).

You might be tempted to conclude that O_F is {a+b√d | a,b∈ℤ}, no matter what d is. But that's not true either. For example, if d=5, you can easily check that α=(1+√5)/2 satisfies x²-x-1=0, so α is an algebraic integer of ℚ(√5).

What actually is true is that O_F = {a+b√d | a,b∈ℤ} if d is square-free and d ≡ 2 or 3 (mod 4). However, if d ≡ 1 (mod 4), then O_F is {(a+b√d)/2 | a,b∈ℤ}. The proof isn't hard, and we'll come back to it later.

In the next installment we'll look further into some of the ring theory relevant to rings of integers.

Tags: algebraic number theory, algebraic integers

Modular arithmetic

After some hiatus, let's return to our discussion of algebraic number theory. Check here for previous articles.

We have already talked briefly about groups and rings (here), but because ring theory is so important to the whole subject, we need to go into it a lot deeper. As preparation for that, we'll look at some of the simplest examples, which will turn out to be greatly generalized in the sequel.

The simplest example of all is probably the ring ℤ of ("rational") integers. We'll assume the properties of ℤ are well enough known as to require no further comment.

The next simplest example is a construction based on ℤ – the ring of integers "modulo" some positive integer n>1. You can read about it in a little more detail here. Of course, if you've studied any elementary number theory, you probably need no further explanation. On the other hand, if your only exposure to it has been courtesy of some writer who talks down to his readers by referring to the topic as "clock arithmetic", the chances are fair to good you could use a less condescending refresher.

In any case, the construction is so important and pervasive in more advanced ring theory and algebraic number theory that it's worth looking at from several points of view. The basic idea could hardly be much simpler. It involves a slightly generalized notion of "equality", or "equivalence" as it's more often referred to.

To begin with, pick and fix an integer n. n is usually assumed positive, though that's not strictly necessary. n, however, does need to be other than 0 or ±1, for reasons which will become obvious. We then say that any other two integers x and y at all are "equivalent modulo n" just in case the difference x-y is divisible by n. (This is why we want n≠0, since division by 0 is meaningless, and also n≠±1, since in that case we would have all integers equivalent modulo ±1. Although that case is meaningful, it isn't very interesting.)

Symbolically, we write x≡y (mod n) if x and y are equivalent modulo n. This notation is chosen to emphasize how ≡ is just a slight variation of the notion of equality. From this point of view, we are still talking about perfectly ordinary rational integers, but using a different notion of when they are "equal". All of the usual facts from elementay number theory about modular arithmetic can be developed from this point of view and the given definition.

But there are other points of view. For example, ≡ (mod n) is just a special case of what is called an "equivalence relation" in set theory. Specifically, let S be any (non-empty) set at all. A relation on a set S is, formally, a mapping from the set of ordered pairs (x,y) of elements of S to the set {0,1} of two elements. If the relation in question is denoted by R, then this function R(x,y) has the value 1 if x and y are in the relation R (which might be, for example, parent and child if S is a set of people). In this case we write xRy, which is not necessarily the same as yRx. On the other hand R(x,y)=0 just in case x and y are not in the indicated relation.

Given all that, there is a special type of relation R on a set S which is technically known as an "equivalence relation". For R to be an equivalence relation, three conditions must be satisfied. First, it must be "reflexive" – xRx for all x∈S. Second, it must be "symmetric" – xRy if and only if yRx, for all x,y. And third, it must be "transitive" – if xRy and yRz then xRz, for all x, y, z.

The interesting thing about equivalence relations is that if R is one on any set S, it is not hard to show that it causes S to be partitioned into disjoint (i. e. non-overlapping) subsets, called "equivalence classes". The classes need not in general all be the same size (regardless of whether S is finite or infinite), but every element of S is in one and only one class, possibly all by itself.

This exercise in abstraction actually has a useful point. Namely, if S=ℤ and R is ≡ (mod n), then obviously we have an equivalence relation on ℤ. Consequently, ℤ is partitioned into equivalence classes. Because of the structure of ℤ and the nature of ≡ (mod n), it turns out that all the equivalence classes are the same size – countably infinite, just like ℤ

From this new point of view, we can give the equivalence classes themselves the structure of a ring. And this ring will be finite in size, having exactly n equivalence classes. We will use the notation ℤ/nℤ for this set of equivalence classes – the reason for the notation will gradually make more sense.

Use the notation [x] for the equivalence class of x∈ℤ. To make a ring out of these equivalence classes, we need to define addition and multiplication. This is easily done: let [x]+[y]=[x+y] and [x][y]=[xy]. There is something subtle that needs to be proved, namely that these definitions are "well-defined", and they do not depend on the choice of a representative element for each equivalence class. For example, suppose [x]=[x′] and [y]=[y′], meaning, if we return to the basic definitions, that x≡x′ (mod n) and y≡y′ (mod n). Then it has to be shown that [x′+y′]=[x+y] and [x′][y′]=[xy]. (This is an easy exercise of unwinding the definitions for the reader.)

Now we can change the point of view just one more time. Let nℤ denote the set of all multiples of integers by the number n. We will redefine x≡y (mod n) from x-y is divisible by n to x-y is a member of the set nℤ. Clearly this doesn't really change anything. The equivalence relation remains the same.

So what have we accomplished? Simply this: starting from the ring ℤ we have found a way to define a new, finite ring ℤ/nℤ, consisting of equivalence classes. In the next installment of this series we will see how to generalize this construction to any "ring of integers of an algebraic number field" in place of ℤ. In the generalization we will replace nℤ with a special type of subring, called an ideal, of the ring of integers.

We will see that the theory of algebraic numbers is largely about the properties of these "ideals". For example, it is not true that any integer of an algebraic number field can be written as a unique product of "prime" numbers. Nevertheless, it is true that any ideal can be written uniquely as a product of "prime" ideals, suitably defined. For many applications in number theory, this is sufficient.

This point of view will also open up many interesting questions. For example, prime ideals in one ring of integers may actually factor into products of distinct prime ideals in the ring of integers of an extension field. And a very central question concerns the rules that govern such factorizations.

Tags: algebraic number theory, modular arithmetic, ring theory

Carnival of Mathematics, Ordinal 5

This edition of the Carnival of Mathematics is dedicated to the memory of Paul J. Cohen (April 2, 1934 - March 23, 2007).

Ironically, he died just 2 weeks ago, the date of the previous edition of this Carnival. Here's the New York Times obituary. There have already been a number of comments about and tributes to Paul in math blogs. One of the shortest, but most significant is from fellow Field's Medalist Terence Tao, who points out that Paul excelled not only in the area of set theory (in which his work best known to the public was done), but also in harmonic analysis.

In fact, Paul was one of the most universal mathematicians (a lot like Tao himself) of the last 50 years. I can personally vouch for that, because he was a Professor in the math department where I did my graduate work. He taught the first year graduate course in complex analysis, and made the subject absolutely inspiring, even though it was new to me at the time, and I was more into algebra than analysis. A year or two later, he conducted a seminar on class field theory (a form of advanced algebraic number theory), and was equally inspiring there. I had the chance to see him in action also in courses or seminars on such subjects as analytic number theory and quadratic forms.

But one thing that impressed me just as much was what a decent, friendly, approachable person he was. Unlike a number of other high-powered mathematicians on the faculty at that time, who were pompous and overbearing. (I won't name names.)

Here's a tribute from another logician, Barkley Rosser, who gives a semi-technical account of the work for which Paul received the Fields Medal. And Jason Rosenhouse at Evolutionblog gives another semi-technical explanation of Cohen's work on the independence from ZFC set theory of the Axiom of Choice and the Continuum Hypothesis.

Paul's set theory work actually settled one of the famous Hilbert Problems, enuciated by David Hilbert in 1900 – the very first, in fact, concerning Georg Cantor's question about the cardinality of the continuum. Was it ℵ₁, or indeed any of the ℵs? Kurt Gödel had earlier shown that the answer was "maybe". What Cohen showed was that the answer was "maybe not". It is a measure of how bright this guy was that set theory and logic were not even specialties of his at the time he took on the problem. According to Ben Yandell's book on Hilbert's problems (The Honors Class), Cohen had no training as a logician, but in 1959 he was looking for a challenging new problem to work on, asked logician Solomon Fefferman for some suggestions, and was told about the continuum problem. By 1963 Cohen had cracked the problem in an entirely original way.

Moving on. There has been a lot more news about mathematicians and mathematics that has appeared in the public media recently – somewhat of an anomaly. One item that received relatively light (but non-null) coverage, because of its very technical nature, is the solution of a problem concerning "mock thera functions", originating in some cryptic notes of another mathematical prodigy, Srinivasa Ramanujan. You can read accounts of this work here, here, here, here, and here. Although I haven't come across any blog articles yet that go into much more detail on this, there is a good, detailed post about the fascinating Ramanujan himself from M. Balamurugan's blog.

Another story that got a bit more attention was the discovery that some ornamental art found on Islamic architecture of the 13^th century (CE) and later has striking affinities to "quasicrystals" and "Penrose tilings". Two of the better articles were done by Philip Ball and Julie Rehmeyer. More stories about this can be found here, here, here, here, here, here, and here. But the most fascinating aspect of this to me is not the Penrose connection, but instead the connection with "noncommutative geometry", as explained by Masoud Khalkhali at (what else?) the Noncommutative Geometry blog.

Of course, the other big piece of mathematical news recently was the determination of the structure of the exceptional (a technical term, not an encomium) Lie group E₈. There have been a number of news reports and blog reports about the work, but this one is interesting, as it comes from a physicist, Clifford Johnson at Asymptotia. That's appropriate, as Clifford is a string theorist, and E₈ plays a big role in string theory, with heterotic strings in particular. Interestingly, Clifford notes that E₈ is also connected with Penrose tilings.

On the other hand, Peter Woit (a mathematician) at Not Even Wrong is not only skeptical of string theory (to put it mildly), but considers all the publicity surrounding E₈ to be a little excessive.

Nevertheless, if you're interested in learning a bit about the mathematics behind E₈, you should certainly take a look at two postings from John Armstrong at The Unapologetic Mathematician – here and here. And if that still isn't enough to satisfy your curiosity, try this wonkier post on Lie groups, Lie algebras, and representations.

By the way, the reason E₈ is called an "exceptional" Lie group is not because of some particularly noteworth properties it possesses. The actual reason is that all Lie groups can be understood in terms of a subclass known as "simple" Lie groups. And most of these, in turn, can be classified in terms of several different infinite subclasses. But there are five other simple Lie groups that defy classification – the exceptional Lie groups. And of these, E₈ is the largest.

Likewise, we have several additional items to mention here that do not fit into a classification scheme.

If you're a fan of hard problems, there are few problems harder than understanding the Navier-Stokes equations. But we can thank Fields medalist Terry Tao for a lucid explanation of
Why global regularity for Navier-Stokes is hard.

If you're more into logical puzzles and computer science, you know all about the "halting problem". Alexandre Borovik at Mathematics under the Microscope uses that topic to illustrate the difference between formal and informal proofs. And incidentally, if you are fond of multi-disciplinary investigations, Alexandre has a book you can download from his site (which is named after the book) that looks at mathematics from the standpoint of cognitive psychology and neuroscience.

My own contribution to this carnival offers a quick (well, compared to a whole book) overview of field theory and Galois theory. It's one step on the way to examining some of the deeper mysteries of algebraic number theory.

Perhaps you've never quite understood why some folks make such a big deal about set theory and the continuum hypotheses, which Paul Cohen did so much to clarify. One illustration of how set theory can actually be applied is to a whole new and elegant way of "constructing" the real numbers. This construction, known as "surreal numbers", was invented by John Horton Conway, another "universal" mathematician. He's also known for, among other things, inventing the "game of life" (based on cellular automata), his role in classifying finite simple groups (only distantly related to Lie groups), and his work (with logician Simon Kochen) on the "free will theorem" in quantum mechanics. Mark Chu-Carroll at Good Math, Bad Math gives us a nice overview of surreal numbers.

Or perhaps you were somehow involved in the "New Math" debacle back in the 70s, and now just don't really care that much for even attempting to explain set theory to junior high students, or practically anyone else for that matter. That's too bad, but math teacher JD2718 has some thoughts on the subject.

In spite of all that, there are rewards to trying to teach math. Dave Marain at MathNotations gives a quick review of the recently published book Coincidence, Chaos, and All That Math Jazz given to him in gratitude by one of his top students. He liked the book.

Thanks, everyone, for reading. Come back again in 2 weeks for the next Carnival of Mathematics, to be hosted by Graeme Taylor at Modulo Errors on April 20.

Note: I had a problem with one of my mailboxes. If you tried to submit an article for this edition of the carnival by sending email to cgd at scienceandreason.net, and the article isn't included here, it may have been affected by the problem. Please resend to carnival at scienceandreason.net, or else submit it for the next edition of the carnival. My apologies for any inconvenience.

Note 2: In the comments Mikael Johanssons observes that his contribution to the carnival got lost in my email problem, so please be sure to check it out. It's about modular representation theory. I'm very sorry about the glitch, Mikael.

Fields and Galois theory

Here's the next installment of our series on algebraic number theory. In the last installment we had a quick look at groups and rings. Now it's time to look at field theory, with special emphasis on what is known as Galois theory. The latter is all about developing a concise description of the relations among the roots of an irreducible polynomial equation using group theory. Some of this theory was famously sketched out in 1832 by Évariste Galois on the night before a duel in which he died.

Galois theory makes it possible to prove several well-known results, such as the impossibility of expressing the solution of some fifth degree polynomial equations in terms of radicals and the impossibility of trisecting some angles with straightedge and compass. We won't go into that, but instead we will eventually see Galois theory used frequently in algebraic number theory.

A field is simply a ring whose multiplication is commutative, has an identity element, and has multiplicative inverses for all elements except the additive identity element. We've already mentioned several examples of fields, specifically number fields, which are algebraic extensions of finite degree of the rationals Q. (I. e., each element of a such a field is an algebraic number in some finite extension of Q.) More exotic examples of fields certainly exist, though, such as finite fields, fields of functions of various kinds, p-adic number fields, and certain other types of local fields. If you go far enough in algebraic number theory, you'll encounter all of these.

The most important set of facts about fields for our purposes lie in what is known as Galois theory. This is the theory developed originally by Évariste Galois to deal (among other things) with the solvability or non-solvability, using radicals, of algebraic equations. It tells us a lot about the structure of field extensions in terms of certain groups – called Galois groups – which are constructed using permutations of roots of a polynomial which determines the extension. (Permutations are 1-to-1 mappings of a set to itself that interchange elements.) A little more precisely, a Galois group consists of automorphisms of a field – i. e. maps (functions) of the field to itself which preserve the field structure. All such automorphisms, it turns out, can be derived from permutations of the roots of a polynomial – under the right conditions.

The importance of Galois theory is that it sketches out some of the "easy" background facts about a given field extension, into which some of the more difficult facts about the algebraic integers of the extension must fit.

Before we proceed, let's review some notations and definitions that will be used frequently. Suppose F is a field. For now, we will assume F is a subset of the complex numbers C, but not necessarily a subset of the real numbers R. If x is an indeterminate (an "unknown"), then F[x] is the set of polynomials in powers of x with coefficients in F. F[x] is obviously a ring. If f(x)∈F[x] is a polynomial, it has degree n if n is the highest power of x in the polynomial. f(x) is monic if the coefficient of its highest power of x is 1. If f(x) has degree n, it is said to be irreducible over F if it is not the product of two (or more) nonconstant polynomials in F[x] having degree less than n.

A complex number α, which is not in F, is algebraic over F if f(α)=0 for some f(x)∈F[x]. f(x) is said to be a minimal polynomial for α over F if f(x) is monic, f(α)=0, and no polynomial g(x) whose degree is less than that of f(x) has g(α)=0. (Note that any polynomial such that f(α)=0 can be made monic without changing its degree.) A minimal polynomial is therefore irreducible over F. F(α) is defined to be the set of all quotients g(α)/h(α) where g(x) and h(x) are in F[x] and h(α)≠0. F(α) is obviously a field, and it is referred to as the field obtained by adjoining α to F.

If E is any field that contains F, such as F(α), the degree of E over F, written [E:F], is the dimension of E as a vector space over F. (Usually this is assumed to be finite, but there are infinite dimensional extensions also.) It is relatively easily proven that if α is algebraic over F and if the minimal polynomial of α has degree n, then [F(α):F]=n. Of course, more than one element can be adjoined to form an extension. For instance, with two elements α and β we write F(α,β), which means (F(α))(β). (Or (F(β))(α) – the order doesn't matter.)

We will frequently need one more important fact. Suppose we have two successive extensions, involving three fields, say D⊇E⊇F. This is called a tower of fields. Then D is a vector space over E, as is E over F. From basic linear algebra, D is also a vector space over F, and vector space dimensions multiply. Consequently, in this situation we have the rule that degrees of field extensions multiply in towers: [D:F]=[D:E][E:F].

Now we're almost ready to define a group, called the Galois group, corresponding to an extension field E⊇F. However, Galois groups can't be properly defined for all field extensions E⊇F. The extension must have a certain property. Here is the problem: The group we want should be a group of permutations on a certain set – the set of all roots of a polynomial equation. But consider this equation: x³-2=0. One root of this equation is the (real) cube root of 2, 2^1/3. The other two roots are ω2^1/3 and ω²2^1/3 where ω=(-1+√-3)/2. You can check that ω³=1 and ω satisfies the second degree equation x²+x+1=0. ω is called a root of unity, a cube root of unity in particular. (Roots of unity, as we'll see, are very important in algebraic number theory.) Now, the extension field E=Q(2^1/3) is contained in R, but the other roots of x³-2=0 are complex, so not in the extension E. This means that it isn't possible to find an automorphism of E which permutes the roots of the equation. Hence we can't have the Galois group we need for an extension like E.

The property of an extension E⊇F that we need to have is that for any polynomial f(x)∈F[x] which is irreducible (has no nontrivial factors) over F, if f(x) has one root in E, then all of its roots are in E, and so f(x) splits completely in E, i. e. f(x) splits into linear (first degree) factors in E. An equivalent condition (as it turns out), though seemingly weaker, is that there be even one irreducible f(x)∈F[x] such that f(x) splits completely in E but in no subfield of E. That is, E must be the smallest field containing F in which the irreducible polynomial f(x)∈F[x] splits completely. E is said to be a splitting field of f(x). The factorization can be written

f(x) = ∏_1≤i≤n (x - α_i)

with all α_i∈E, where n is the degree of f(x). (Remember that we are assuming f(x) is monic.) When this is the case, E is generated over F by adjoining all the roots of f(x) to F. In this case it can be shown that the degree [E:F] is the same as the degree of f(x).

An extension that satisfies these conditions is said to be a Galois extension, and it is the kind of extension we need in order to define the Galois group G(E/F). (Sometimes the type of extension just described is called a normal extension, and a further property known as separability is required for a Galois extension. As long as we are dealing with subfields of C, fields are automaticaly separable, so the concepts of Galois and normal are the same in this case.)

Suppose E⊇F isn't a Galois extension. If E is a proper extensions of F (i. e. E≠F), if α∈E but α∉F, and if f(x) is a minimal polynomial for α over F, then the degree [E:F] of the extension is greater than or equal to the degree of f(x). The degrees might not be equal, because all the roots of f(x) must be adjoined to F to obtain a Galois extension, not just a single root. If α is (any) one of the roots, [F(α):F] is equal to the degree of f(x). But this is the degree [E:F] only if α happens to be a primitive element for the extension, so that E=F(α), which isn't usually the case, and certainly isn't if E isn't a Galois extension of F.

In the example above with f(x)=x³-2, we have E = Q(ω,2^1/3) = Q(ω)(2^1/3), [Q(ω):Q]=2 and [Q(ω,2^1/3):Q(ω)]=3, so the degree of the splitting field of f(x) over Q is 6, because degrees multiply. Q(2^1/3)⊇Q is an example of a field extension that is not Galois. But Q(ω,2^1/3)⊇Q(ω) is Galois, since f(x) is irreducible over Q(ω) but splits completely in the larger field. Likewise, Q(ω)⊇Q is Galois, and in fact all extensions of degree 2 are Galois. (If f(x)∈Z[X] is a quadratic which is irreducible over Q and has one root in E, then the roots are given by the quadratic formula and involve √d for some d∈Z, so if one is in E, both are.)

We'll come back to this example, but first we'll look at a simpler one to get some idea of how Galois groups work. Consider the two equations x²-2=0 and x²-3=0. The roots of the first are x=±√2, and the roots of the second are x=±√3. We will start from the field Q and adjoin one root of each equation. This yields two different fields: E₂=Q(√2) and E₃=Q(√3). If we adjoin a root from both equations we get a larger field that contains the others as subfields: E=Q(√2,√3).

Consider the field extension E₂⊇Q first. We use the notation G(E₂/Q) to denote the Galois group of the extension. In this example, call it G₂ for short. We will use Greek letters σ and τ to denote Galois group elements in general. G₂ consists of two elements. One of these is the identity (which we denote by "1") which acts on elements of the field E₂ but (by definition) leaves them unchanged. This can be symbolized as 1(α)=α for all α∈E₂. The action of a Galois group element can be fully determined by how it acts on a generator of the field, meaning √2 in this case. So it is enough to specify that 1(√2) = √2. This Galois group has just one other element σ₂, which is defined by σ₂(√2)=-√2. An important property that a Galois group must satisfy is that the action of all its elements leaves the base field (Q in this case) unchanged. A Galois group is an example of a group that acts on a set – a very important concept in group theory. But there is an additional requirement on Galois groups: each group element must preserve the structure of the field it acts on. In technical terms, it must be a field automorphism. We'll see the importance of this condition very soon.

As you can probably anticipate, the Galois group G₃=G(E₃/Q) has elements 1 and σ₃ defined by σ₃(√3)=-√3. We can now ask: what is the Galois group of the larger extension E⊇Q? It must contain 1, σ₂ and σ₃. We have to think about how (for instance) σ₂ acts on √3. The clever thing about Galois theory is that it's easy to say what this action should be: σ₂ should leave √3 unchanged: σ₂(√3)=√3. In particular, σ₂(√3) cannot be ±√2 The reason is that σ₂ leaves the coefficients of x²-3=0 unchanged, and because σ₂ is a structure-preserving field automorphism it cannot map something that is a root of that equation (such as √3) to something that is not a root of that equation (±√2).

For any finite group G, the order of the group is the number of distinct elements. We symbolize the order of G by #(G). In Galois theory it is shown that the order of a Galois group is the same as the degree of the corresponding field extension. Symbolically: #(G(E/F))=[E:F]. Basically this is because we can always find a primitive element θ such that E=F(θ), and θ satisfies an equation f(x)=0, where the degree of f(x) is [E:F]. The other n-1 roots of that equation are said to be conjugate roots. We get n automorphisms, the elements of G(E/F), generated from mapping θ to one of its conjugates (or to itself, giving the identity automorphism). Since the degrees of field extensions in towers multiply, so too do the orders of Galois groups in field towers, as long as each extension is Galois. That is, if D⊇E⊇F, where each extension is Galois, then #(G(D/F)) = #(G(D/E))#(G(E/F)). In our example, the degree of the extension is [Q(√2,√3):Q] = [Q(√2,√3):Q(√2)][Q(√2):Q] = 4. So this is also the order of the Galois group G=G(Q(√2,√3)/Q), and therefore we need to find 4 elements.

We've already identified three of the elements (1, σ₂ and σ₃). It's pretty clear that the remaining element must be a product of group elements: τ=σ₂σ₃. The product of Galois group elements is just the composition of the elements, which are field automorphisms (which happen to be derived from permutations on roots of equations), and hence they compose like any other function (or permutation). (Composition is just another term for the the function which is the result of applying one function after another.) Because of how σ₂ and σ₃ are defined, it must be the case that τ(√2)=-√2 and τ(√3)=-√3. Since E⊇Q is generated by √2 and √3, and τ is a field automorphism, we can figure out what τ(α) must be for any other α∈E. For instance, τ(√6)=√6, since √6=√2√3.

(Remember that we specified σ₂(√3)=√3. You may have been wondering why we didn't just define the action of σ₂ as an element of the full Galois group G=G(E/Q) by σ₂(√3)=-√3. Had we done that, σ₂ would have been what we found as τ, while the τ we got as the product of σ₂ and σ₃ would turn out to be the "old" σ₂, so the only difference would be a relabeling of group elements.)

For a slightly more complicated example, suppose f(x)=x²+x+1 and g(x)=x³-2, with roots ω and 2^1/3 respectively, as above. Then in the tower Q(ω,2^1/3) ⊇ Q(ω) ⊇ Q both the extensions are Galois. (We already saw this isn't so with the tower Q(ω,2^1/3) ⊇ Q(2^1/3) ⊇ Q – order matters.) So the full extension E=Q(ω,2^1/3) ⊇ Q is Galois. Its Galois group G=G(E/Q) has order 6, because 6 is the degree of the whole extension, since the intermediate extensions are of degree 3 and 2 and the degrees of the extensions multiply.

It turns out to be easy to determine the Galois group of this extension, although there are some tedious calculations needed to verify this. So bear with us a moment here. We can define two automorphisms of E that leave Q fixed, as follows. It suffices to specify them on generators of the field. Let one automorphism σ be defined by σ(&omega)=ω² and σ(2^1/3)=2^1/3. Let the other automorphism τ be defined by &tau(2^1/3)=ω2^1/3 and τ(ω)=ω. σ and τ are defined to leave elements of Q unchanged. For sums and products elements of E, σ and τ are defined to preserve the field structure, so they really are automorphisms (though, to be rigorous, this should be checked). So σ and τ are elements of the Galois group G=G(E/Q).

We can also see that σ²(ω) = σ(σ(ω)) = σ(ω²) = ω⁴ = ω, because ω³ = 1. So σ² is the identity automorphism. (Note that the exponents on σ and τ refer to repeated composition, not ordinary exponentiation, because composition "is" multiplication in the group G.) If we compute τ² and τ³ in the same way, applied to 2^1/3, we find that τ²(2^1/3) = ω²2^1/3, and τ³(2^1/3) = 2^1/3, again because ω³ = 1. Thus τ² isn't the identity automorphism, but τ³ is.

Now let's compute with the composed automorphisms στ and τσ. First, στ(2^1/3) = σ(ω2^1/3) = ω²2^1/3. However, τσ(2^1/3) = τ(2^1/3) = ω2^1/3. So we have στ ≠ τσ, because ω≠ω². Instead, we will find by a similar calculation that στ(2^1/3) = ω²2^1/3 = τ²σ(2^1/3). Hence στ = τ²σ. A little more checking will show that 1 (the identity automorphism), σ, τ, τ², τσ, and στ give a complete list of distinct automorphisms that can be formed from σ and τ. That's just right, because G must be a group of order 6.

In abstract group theory there are only two distinct groups of order 6. (That is, distinct up to an isomorphism, which is a 1-to-1 structure-preserving map between groups that shows they are essentiall the "same" group.) One is the cyclic group of order 6, denoted by C₆. This is isomorphic to the direct product of a cyclic group of order two and one of order 3, i. e. the group C₂×C₃. However, since στ ≠ τσ, G isn't abelian, it cannot be C₆, which is abelian. The only other group of order 6 is (up to isomorphism) S₃, the group of permutations of three distinct objects, also known as the symmetric group. (An isomorphic group is the dihedral group D₃, the group of symmetries of an equilateral triangle.) Since this group is the only nonabelian group of order 6, G(E/Q) must be isomorphic to it.

There's a whole lot more that could be said about Galois theory, but that would take up quite a bit of space, and the intention here is only to give a feel for what it is about. The basic idea to take away is this: A great deal is known about abstract groups and their subgroup structure. Galois theory is a way to "map" extensions of fields to groups and their subgroups in such a way that most of the interesting details about the extension are reflected in details about the groups, and vice versa. The group structure is sensitive to relationships among elements in the subextensions of a Galois extension. In Galois theory it is proven that there is a precise correspondence between subextensions and subgroups of the Galois group.

It thus becomes possible to infer facts about field extensions easily from a knowledge of their Galois groups. One example of the power of this method is that it made possible proving facts that had remained mysterious for hundreds of years – for example, the unsolvability by radicals of general polynomial equations of degree 5 or more, and the impossibility of certain geometric constructions by straightedge and compass alone (trisecting angles, for example).

Galois theory is an absolutely indispensible tool in algebraic number theory. It will come up again and again. We will mention other results in the theory when they are needed.

In the next installment we'll circle back to take a deeper look at ring theory, which is the most basic tool used in algebraic number theory – because there are generalizations of "integers" in an algebraic number field, and they are rings analogous to the familiar ring Z of ordinary integers.

Tags: algebraic number theory, field theory, Galois theory, Galois group

E8

Hi there, mathematicians. I assume you've heard the big news about E₈ by now:

248-dimension maths puzzle solved

An international team of mathematicians has detailed a vast complex numerical "structure" which was invented more than a century ago.

Mapping the 248-dimensional structure, called E8, took four years of work and produced more data than the Human Genome Project, researchers said.

Lie groups aren't a specialty of mine, so I'll hold off writing about this (for now), but you might be interested to read what a few other are saying.

• AIM press release: Mathematicians Map E₈
  
• New York Times: The Scientific Promise of Perfect Symmetry
  
• Good Math, Bad Math: The Mapping of the E₈ Lie Group (Minor Update)
  
• Not Even Wrong: E₈ Media Blitz
  
• The n-Category Café: News about E₈
  
• The Unapologetic Mathematician: A rough overview
  
• The Unapologetic Mathematician: Lie groups, Lie algebras, and representations
  

Tags: E₈, Lie groups, string theory

Groups and rings

In our previous installment of the series on algebraic number theory, we took a little detour into Diophantine equations in order to provide some motivation for the theory itself. Prior to that, we had looked at different types of numbers, to give a perspective on the sorts of objects the theory deals with.

In those discussions, we touched (explicitly or implicitly) on abstract algebraic structures called groups, rings, and fields. All three of these concepts are incredibly important in the theory of algebraic numbers, and to a very large extent in the rest of modern mathematics as well. Today we'll deal with groups and rings. In the next installment we'll take up fields and Galois theory. This won't be in any great detail – just the basic concepts. Further depth will be introduced later, as it becomes necessary.

Groups

We begin with groups. As with most other sorts of algebraic systems, groups are defined abstractly in terms of sets of elements satisfying certain axioms. The axioms for a group are not the simplest that an interesting mathematical system can have -- monoids and semigroups have somewhat weaker axioms. But groups are just about the simplest objects that occur commonly in algebraic number theory.

A group G is a mathematical system consisting of a set of elements and one operation between any two elements of the set. If "∘" denotes the operation, in a group there are three requirements:

1. the operation should be associative: x∘(y∘z) = (x∘y)∘z for all x, y, z in G;
   
2. there should be an identity element "e": e∘x = x∘e = x for all x in G;
   
3. every element of G should have an inverse: x∘x^-1 = x^-1∘x = e.
   

These axioms can be stated in slightly different ways, but we don't need to get into that.

Note one respect in which these group rules are different from the usual rules of either addition or multiplication in arithmetic: the commutative property x∘y = y∘x is not required for elements of a group, though it might hold for some or even all elements. If it does hold for all group elements, the group is said to be commutative or abelian (after Niels Abel). In the theory of algebraic numbers, whenever groups consist of actual algebraic numbers they will necessarily be commutative, since the rules of arithmentic (both addition and multiplication) still hold. But we will encounter groups that are defined in different ways that definitely won't be commutative. Some of the hardest problems of the theory, in fact, occur in the non-commutative cases.

For a nontrivial example of a commutative group that's important in algebraic number theory, just look at the set of all units, as we discussed in reference to Pell's equation. As you recall, we denoted by Z[√n] the set of numbers of the form a+b√n, where a and b are integers, and n is a positive integer that's not a perfect square. (Z[√n] is in fact a ring, as we'll define the term in a moment.)

Within that set, consider the subset of numbers such that the equation a² - nb² = ±1 holds. In other words, the "norm" of a+b√n, N(a+b√n), as defined by the left hand side of the equation, has the value ±1. We noted that this condition is necessary and sufficient for a number in the subset to have a multiplicative inverse. We called such numbers units of the ring Z[√n]. Note that 0 is not a unit, but 1 is, and that the existence of a multiplicative inverse of any unit makes the set of units into a commutative group under multiplication (with ordinary addition being irrelevant in this group – indeed, the sums and differences of units are not units).

Another thing to note is that the requirement for an identity element is a requirement for a solution to a certain simple equation, and we have seen this in action several times. For instance, the natural numbers N (nonzero integers) do not form a group under addition, because there is in general no solution to an equation of the form x+a = 0 with arbitrary a∈N. But if we extend N to the integers Z by "adjoining" all negative numbers, we have in effect simply included all formal solutions of equations x+a=0 for each a∈N and gotten lucky in that the enlarged domain satisfies the group axioms without difficulty.

Very much the same thing happened with respect to the operation of multiplication when we passed from the integers Z to the rationals Q. Again, with respect to multiplication, Z satisfies the group axioms except for the existence of inverses. That is, we are not able to solve the equation xa = 1 for arbitrary a∈Z. In fact, a solution exists only for a = ± 1. But in defining the rational numbers Q in effect we just formally adjoined the inverses (reciprocals) 1/a for each a∈Z (except a=0). The resulting group with the operation of multiplication consists of all nonzero elements of Q. This group is sometimes denoted by Q^×.

If we wanted to preserve the additive structure of Z at the same time as providing multiplicative inverses, in order to construct the ring Q, we would need to have been a little subtler. This process is a standard one. It is called constructing a field of fractions, and we will come back to it.

Rings

Let's look at rings next, since they are the next major level up in axiom complexity. A ring is a mathematical system which has two distinct binary operations: "+" and "×", which are intended to be rather like the addition and multiplication of ordinary arithmetic. If R is a ring, then it satisfies the axioms for a commutative group with respect to addition. With respect to multiplication, R must satisfy the associative axiom. Sometimes rings are not required to have a multiplicative identity element, but most in fact do. Inverse elements, however, do not typically exist, even if there is a multiplicative identity element. Addition in a ring is always commutative, but multiplication need not be. If the multiplication is commutitive, the ring is a commutative ring. The rings that occur in algebraic number theory are commutative rings if they consist of ordinary algebraic numbers, but a few important cases of rings (matrix rings for example) aren't commutative.

In addition to the requirements on the operations of addition and multiplication seprately, they must satisfy a compatibility condition, known as the distributive law of multiplication with respect to addition:

• for all a, b, c in R, a(b + c) = ab + ac
  

If multiplication in R isn't commutative, the same thing must hold for multiplication on the right as well. One consequence of this axiom is that if 0 is the additive identity element, a0=0a=0 for all a∈R.

The integers Z are the most obvious example of a ring. For any field F containing Q, there is also the concept of a ring of integers of the field F. This ring is a direct generalization of Z, and it is one of the central objects of study. One wants to know as much as possible about the structure of such rings, because this knowledge has extensive practical application to the study of Diophantine equations, as we shall see. Elements of a such a ring of integers are called, simply, algebraic integers.

In many respects, of groups, rings, and fields, it is rings which are most interesting. They have the complexity due to possessing two operations, but the freedom of a less restrictive set of axioms than fields. This results in many more special situations, though not all of the strong theorems about fields (such as Galois theory) apply to rings.

We'll have a lot more to say about abstract ring theory, but in the next installment the theory of fields and Galois theory will be reviewed.

Tags: algebraic number theory, group theory, ring theory