Cyclotomic fields, part 2

In our previous article on cyclotomic fields we were talking about why the Galois group G of ℚ(μ_n)/ℚ is isomorphic to (ℤ/nℤ)^×, where n∈ℤ and μ_n is the group of n^th roots of unity, the roots of xⁿ-1=0 in some extension of ℚ. (Check here for a list of previous articles on algebraic number theory.)

So far we've shown that G is isomorphic to a subgroup of (ℤ/nℤ)^×. We still need to show it is actually isomorphic to the whole group, or equivalently that |G|, the order of G, which is equal to the degree of the field extension, [ℚ(μ_n):ℚ], actually equals |(ℤ/nℤ)^×|, which is φ(n), the number of positive integers less than and relatively prime to n.

The group μ_n is cyclic. Any generator of the group is, by definition, a primitive n^th root of unity. We let ζ be an arbitrary but fixed such generator. Then ℚ(μ_n)=ℚ(ζ). Let f(x) be the minimal polynomial of ζ. f(x)∈ℚ[x] is irreducible over ℚ. All other elements of μ_n are of the form ζ^a for some a∈ℤ, where a is well-defined modulo n. Further, ζ^a is a primitive n^th root of unity if and only if a is relatively prime to n, i. e. the greatest common divisor (a,n)=1. The degree of f(x) equals [ℚ(μ_n):ℚ] and |G|. At this point, all we know about these numbers is that they divide φ(n) (since |G| does).

The group homomorphism j:G→(ℤ/nℤ)^× was defined by the relation σ(ζ)=ζ^j(σ), for σ∈G. We showed that j(σ) is injective and independent of the choice of ζ. Hence G is isomorphic to a subgroup of (ℤ/nℤ)^×, and therefore the degree of f(x) (and [ℚ(μ_n):ℚ] and |G|) is ≤φ(n). G is isomorphic to a proper subgroup of (ℤ/nℤ)^×, i. e. not the whole group, if j is not surjective and the degree of f(x) is strictly less than φ(n).

As we noted last time, the problem here is that we don't yet know that all φ(n) primitive n^th roots of unity are zeroes of f(x), so that |G| and the degree of f(x) equal φ(n), and hence G(ℚ(μ_n)/ℚ) ≅ (ℤ/nℤ)^×. Stated another way, we don't yet know that the field homomorphism on ℚ(μ_n) induced by mapping ζ to ζ^a is actually an automorphism of the field, hence an element of the Galois group. It could fail to be if, say, the minimal polynomial of ζ^a is different from that of ζ, which could happen if the degree of f(x) is less than φ(n) because not all primitive n^th roots of unity are zeroes of f(x). In order to rule out this possibility, we will show that the degree of f(x) is ≥φ(n).

There are various ways to prove the isomorphism, and even a number of ways to prove that f(x) has φ(n) distinct roots, so its degree is ≥φ(n). Many of these proofs use machinery (such as discriminants, factorization and ramification of primes, etc.) that we haven't extensively discussed yet, so I'll avoid using such things in the proof. However, we'll get to these topics eventually, and also show a way to construct the automorphisms σ∈G explicitly – after finishing the proof that G(ℚ(μ_n)/ℚ) ≅ (ℤ/nℤ)^×.

So let's get started. The roots of the minimal polynomial f(x)∈ℚ[x] are all conjugates σ(ζ) for σ∈G, so f(x)=Π_σ∈G(x-σ(ζ)). Hence f(x) is monic (leading coefficient 1). The coefficients of f(x) are symmetric functions of all conjugates of ζ, so the coefficients are all left fixed by all σ∈G. f(x) divides xⁿ-1, so all its roots – the conjugates of ζ – are algebraic integers. So the coefficients are also algebraic integers (sums of products of powers of algebraic integers) – members of the ring of integers O_ℚ(ζ). They are also in the base field, since they're left fixed by G. A basic fact is that O_ℚ(ζ)∩ℚ = ℤ – any algebraic integer that lies in the base field is necessarily an integer of the base field. Hence f(x)∈ℤ[x].

Suppose that for any a∈ℤ relatively prime to n, i. e. (a,n)=1, ζ^a is also a root of f(x): f(ζ^a)=0. Since these ζ^a with 1≤a<n are distinct primitive n^th roots of unity if ζ is, and there are φ(n) of them, the degree of f(x), and hence |G|, is ≥ φ(n). But we already showed |G|≤φ(n), hence |G|=φ(n). Since G is isomorphic to a subgroup of (ℤ/nℤ)^×, we must actually have an isomorphism: G ≅ (ℤ/nℤ)^×.

So all we have to show is f(ζ^a)=0 for 1≤a<n and (a,n)=1. The first thing to note is that it suffices to prove this just for primes p with (p,n)=1. For suppose we had that. For general a with (a,n)=1, let p be a prime that divides a. Then (p,n)=1. Consider ζ^a/p. Since (a/p,n)=1, ζ^a/p is a primitive n^th root of unity with one fewer prime divisor in the exponent than ζ^a. So by induction on the number of prime divisors of the exponent f(ζ^a/p)=0. But if the result is true for prime powers of primitive n^th roots of unity that satisfy f(x)=0, then f(ζ^a)=0 since ζ^a=(ζ^a/p)^p. Alternatively, you can recall that (according to a theorem of Dirichlet), there are infinitely many primes p in the arithmetic progression a+nk for (a,n)=1 and k∈ℤ. Since ζⁿ=1, ζ^a=ζ^p for all such p.

So let p be prime and (p,n)=1. Then note that f(x)^p-f(x^p)∈pℤ[x] for any f(x)∈ℤ[x]. This can be proved by induction on the degree of f(x). Suppose the highest degree term of f(x) is Ax^m, with p∤A. Then (Ax^m)^p-Ax^mp∈pℤ[x] because A^p≡A (mod p), because (ℤ/pℤ)^× is cyclic of order p-1 (Fermat's theorem). So if h(x)=f(x)-Axⁿ, then we just have to show h(x)^p-h(x^p)∈pℤ[x]. But that can be assumed true by induction, unless the degree of h(x) is 1. In the latter case, if h(x)=Ax+B, we need (Ax+B)^p-(Ax^p+B)∈pℤ[x]. But all the coefficients in (Ax+B)^p except the first and last contain binomial coefficients divisible by p, and the remaining terms are handled with Fermat's theorem as before.

Finally, then, suppose the opposite of what we want to show, namely that there is a prime p with p∤n and f(ζ^p)≠0. By what we just showed, f(ζ^p) is divisible by p in O_ℚ(ζ). We have f(x)=Π_i∈I(x-ζⁱ) for I={i∈ℤ: 1≤i<n and (i,n)=1}. So f(ζ^p) divides a product of nonzero factors ζ^p-ζⁱ. By a lemma we'll prove in a moment, if J={(i,j): i,j∈ℤ, 0≤i,j<n, i≠j}, Π_(i,j)∈J(ζⁱ-ζ^j) = (-1)^n-1nⁿ. Hence f(ζ^p) divides nⁿ and p|n, contrary to assumption. This contradiction means f(ζ^p)=0, as required. We've now shown f(x) has at least φ(n) roots, hence G(ℚ(μ_n)/ℚ) ≅ (ℤ/nℤ)^×.

Now for the last lemma: We have xⁿ-1=Π_0≤i<n(x-ζⁱ). Equating the constant terms gives (-1)^n-1=Π_0≤i<nζⁱ. And by taking derivatives of both sides, nx^n-1=Σ_0≤i<nΠ_{0≤j<n, j≠i}(x-ζ^j). Substituting x=ζ^k, nζ^k(n-1)=Π_{0≤j<n, j≠k}(ζ^k-ζ^j). Taking products of this for 0≤k<n gives, with the set J as above, Π_(i,j)∈J(ζⁱ-ζ^j) = nⁿ(Π_0≤k<nζ^k)^n-1. But the last product on the right side was evaluated above, so finally we are left with (-1)^n-1nⁿ on the right (since n-1 has the same even/odd parity as its square).

Well, that was a bit of work, wasn't it? But nothing too esoteric, apart from a little Galois theory and some classic number theoretical facts. (Thanks to [1, pp 96-8] for the bulk of the proof.)

Actually, it is possible to do this without the lemma, using the theorem on primes in an arithmetic progression. Suppose f(x) is any polynomial in ℤ[x] such that f(ζ)=0 when ζ is a primitive n^th root of unity. Then for any a∈ℤ with (a,n)=1, since f(x)^p-f(x^p)∈pℤ[x] for any prime p∈ℤ, we have 0=f(ζ)^p≡f(ζ^p) mod pO_ℚ(ζ). But there are infinitely many primes p≡a mod n, and for such p, ζ^p=ζ^a. Consequently, f(ζ^a) is a member of an infinite number of distinct prime ideals, which is possible only if f(ζ^a)=0. Hence f(x) has degree ≥φ(n), which is the crucial fact we found before.

We can now define the cyclotomic polynomial Φ_n(x)=Π_{0<i<n, (i,n)=1}(x-ζⁱ), for any primitive n^th root of unity ζ. From the foregoing, we know a lot about Φ_n(x): its roots are precisely all the primitive n^th roots of unity (in ℂ), its degree is φ(n), it is irreducible (over ℚ), its coefficients are in ℤ, and it is the minimal polynomial of ζ. The notation Φ_n(x) is on account of its relation to the Euler function φ(n).

We also have this factorization of xⁿ-1 in ℤ[x]: xⁿ-1 = Π_d|nΦ_d(x). This holds, since the roots of each Φ_d(x) are precisely the roots of unity in the cyclic group μ_n that have exact order d for each d that divides n. (Each root has one and only one exact order d satisfying d|n.) This relation is occasionally useful, and it yields interesting facts such as Σ_d|nφ(d) = n (by taking degrees of polynomials on both sides).

It turns out that the irreducibility of Φ_n(x) is relatively easy to prove for certain n, namely those that are powers of a single prime. So let p be prime and q=p^r for an integer r≥1. Let f(x)=Φ_q(x). The roots of f(x) are primitive q^th roots of unity, namely ζ∈μ_q such that ζ has order q. There are φ(q) of these and φ(q)=q-q/p=q(1-1/p)=(p-1)p^r-1 (because every p^th element of the set {0,1,...,q-1} is divisible by p). So clearly f(x)=(x^q-1)/(x^q/p-1). Let g(x)=(x^p-1)/(x-1) and h(x)=g(x+1)=((x+1)^p-1)/x=x^p-1+Σ_0<j<p(^p _j)x^j-1, where (^p _j) is a binomial coefficient, which is divisible by p if 0<j<p. Finally, consider the polynomial h(x^q/p)=g(x^q/p+1).

Suppose f(x) splits in ℚ[x]. Then since f(x)=g(x^q/p), the latter splits, and consequently g(x^q/p+1)=h(x^q/p) does too. But h(x^q/p) is what's known as an Eisenstein polynomial, because the leading coefficient is not divisible by p, the constant term is p (not divisible by p²), and all other nonzero coefficients (the binomial coefficients) are divisible by p. However, Eisenstein polynomials are irreducible over ℚ. This contradiction means f(x) must be irreducible over ℚ. QED.

The fact that Φ_q(x) is irreducible if q=p^r, and hence G(ℚ(μ_q)/ℚ)≅(ℤ/qℤ)^×, can be used as the basis for yet another proof of this isomorphism for arbitrary n, by considering prime power divisors q of n, the corresponding extensions ℚ(μ_q)/ℚ, and their Galois groups in building up the full extension ℚ(μ_n)/ℚ and its Galois group. But we won't go into that now.

In the next installment, we'll discuss many more fun facts about cyclotomic fields.

References:

[1] Goldstein, Larry Joel - Analytic Number Theory