## 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 $p = 5.)$