Cyclic extension of degree $p^2$ ramified at $p$ and outside $p$

Question

The main reduction step in proving the Kronecker-Weber Theorem is to reduce the theorem to proving the following result:

Theorem. If $K/\mathbb{Q}$ is a cyclic extension of degree $p^n$ which is unramified outside $p$, then $K \subset \mathbb{Q}(\zeta_{p^{n+1}})$.

My question: Is it possible to construct a cyclic extension of degree $p^n$ (take $n=2$ for convenience) which is ramified both at $p$ and at another prime $q\neq p$? It is clear that such a prime must be tamely ramified, but how does one construct $K$?

Answer 1

Take a prime $q \equiv 1 \bmod p^n$; then the field of $q$-th roots of unity has a cyclic subfield of degree $p^n$. Compose it with the cyclic field ramified exactly at $p$ an look for a suitable cyclic subextension different from those ramified at only one prime.

Answer 2

Actually you can get hold of all the cyclic extensions of the type you are looking for. Things are clearer if we start taking a "higher" point of view. For a number field $K$ and an (odd, for simplification) prime $p$, fix a finite set $S$ of primes of $K$ containing all the primes above $p$, and denote by $G(S)$ the Galois group over $K$ of the maximal pro-$p$-extension of $K$ which is unramified outside $S$ (i.e. the compositum of all the finite $p$-extensions of $K$ with that property), and by $X(S)= G(S)^{ab}$ its maximal abelian quotient. By CFT, the $\mathbf Z_p$-module $X(S)$ is of finite type, of the form $\mathbf Z_p^{r} \times T(S)$, where $T(S)$ is the (finite) $\mathbf Z_p$-torsion. The free factor $\mathbf Z_p^{r}$ is the Galois group over $K$ of the compositum of all the $\mathbf Z_p$-extensions of $K$, i.e. of the Galois extensions of $K$ whose Galois groups are isomorphic to $(\mathbf Z_p , +)$. It is known that such extensions are unramified outside {$p$}, so that for your purpose, we must look at the torsion $T(S)$. Using Galois cohomology, it can be shown that a minimal system of generators of $T(S)$ is obtained by "abelianizing" a minimal system of relations of the pro-$p$-group $G(S)$; see e.g. H. Koch's "Galois theory of $p$-extensions", chapter 11.

The aforementioned relations are generally unknown, but they are explicit (just as the K-W.theorem is explicit) in the case $K=\mathbf Q$. In your situation where $S$={p, q}, $G(S)$ can be described minimally by 2 generators $t_p , t_q$ with 1 single relation of the form $t^{q-1}$ modulo commutators (op. cit., example 11.11). This means in particular that $G(S)$ is a free pro-$p$-group (i.e. $T(S)=0$) iff $q\neq1\mod p$, in which case you cannot get your desired cyclic extension. In the other cases, take $q\equiv1\mod p^n$ as in the answer of @franz lemmermeyer.