Elementary Class Groups Updated

In a previous post, I gave a short argument showing that, for odd primes p and N such that N \equiv -1 \mod p, the p-class group of \mathbf{Q}(N^{1/p}) is non-trivial. This post is just to remark that the same argument works under weaker hypotheses, namely:

Proposition: Assume that N is p-power free and contains a prime factor of the form q \equiv -1 \mod p, and that p is at least 5. Then the p-class group of K = \mathbf{Q}(N^{1/p}) is non-trivial.

The proof is pretty much the same. If N has a prime factor of the form 1 \mod p, then the genus field is non-trivial. Hence we may assume there are no such primes, from which it follows that H^1_S(\mathbf{F}_p) has dimension one and H^2_S(\mathbf{F}_p) is trivial, where S denotes the set of primes dividing Np. The prime q gives rise to a non-trivial class b \in H^1_S(\mathbf{F}_p(-1)) which is totally split at p (this requires that p be at least 5), and the field K itself gives rise to a class a \in H^1_S(\mathbf{F}_p(1)). But now the vanishing of H^2 implies that a \cup b = 0 and hence there exists a representation of G_S of the form:

\rho: G_S \rightarrow \left( \begin{matrix} 1 & a & c \\ 0 & \chi^{-1} & b \\0 & 0 & 1 \end{matrix} \right),

where \chi is the mod-p cyclotomic character. The class c gives the requisite extension (after possibly adjusting by a class in the one-dimensional space H^1_S(\mathbf{F}_p)). The main point is that the image of inertia at primes away from p is tame and so cyclic, but any unipotent element of \mathrm{GL}_3(\mathbf{F}_p) has order p if p is at least three. This ensures c is unramified over K away from the primes above p. On the other hand, the class b is totally split at p. This implies that the class c is locally a homomorphism of the Galois group of \mathbf{Q}_p, and so after modification by a multiple of the cyclotomic class in H^1_S(\mathbf{F}_p) may also be assumed to be unmarried at p. The fact that b \ne 0 ensures that c \ne 0, and moreover the fact that p is at least 5 implies that the kernel of c is distinct from that of a, completing the proof. (This result was conjectured in the paper Class numbers of pure quintic fields by Hirotomo Kobayashi, which proves the claim for p = 5.)

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2 Responses to Elementary Class Groups Updated

  1. Emmanuel Kowalski says:

    Nice typo: “unmarried at p”…

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