Maximal unramified outside $p$ extension contained in $\mathbb Q(\zeta_n)$

Question

Let $n$ be a positive integer. Let $p$ be a prime. and let $n=p^ak$ where $p$ doesn't divide $k$. Is the subfield $\mathbb Q(\zeta_{p^a})$ of $\mathbb Q(\zeta_n)$ maximal among subfields of $\mathbb Q(\zeta_n)$ which are unramified outside $p$?

Answer 1

Yes. Any proper extension field $L$ of $\mathbb{Q}(\zeta_{p^a})$ that is contained in $\mathbb{Q}(\zeta_n)$ can be written as the compositum $K \mathbb{Q}(\zeta_{p^a})$ with $K = L \cap \mathbb{Q}(\zeta_k)$ and $[K \colon \mathbb{Q}] \geq 2$. Since $\mathbb{Q}$ admits no unramified extensions, some prime $q$ ramifies in $K/\mathbb{Q}$. From the theory of cyclotomic fields, $q$ must divide $k$, so $q \neq p$. Thus, the inertial degree of $q$ in $L/\mathbb{Q}$ is $\geq 2$. Since $q \neq p$, then also $\mathbb{Q}(\zeta_{p^a})/\mathbb{Q}$ is unramified above $q$. Through multiplicativity of inertial degrees, it must happen that the primes above $q$ in $\mathbb{Q}(\zeta_{p^a})$ ramify in $L$, and this answers your question.

When one studies your question in the context of class field theory, it becomes a special case of an interesting more general question: given a number field $F$, what can we say about the maximal Abelian extension $M$ of $F$ that is unramified away from some prime $p$? Work on this question has been done by George Gras, Thong Nguyen Quang Do, Jean Francois-Jaulent, and others. In general, it is easier to work locally, and we split $\mathrm{Gal}(M/F)$ into a profinite $p$-group and a group of order relatively prime to $p$. The latter part is much easier to understand.

The harder part is to understand the maximal Abelian extension of $F$ of $p$-power order that is unramified away from $p$. Here is a special case: if $F$ is a totally real number field (i.e. in every embedding into $\mathbb{C}$, the image of $F$ is in $\mathbb{R}$), then the maximal subfield of $F(\zeta_{p^a})$ of $p$-power degree over $F$ is totally real and unramified away from $p$. It can happen that every such extension of $F$ of $p$-power degree is contained in one of these fields, as is the case with $\mathbb{Q}$ (as your question shows), but other times this is false. What is interesting is that the existence of other such fields is determined by simple invariants of $F$, the main ones being the class group and the $p$-adic regulator. Leopoldt's conjecture states that you can never find an infinite tower of such extensions that is disjoint from the cyclotomic one -- it is still open.