TITLE: discriminant of subfield of $\mathbb{Q}(\zeta_p)$
QUESTION [6 upvotes]: For a field $K\subset \mathbb{Q}(\zeta_p)$ $~$($\zeta_p$ a primitive pth root of unity, p a prime), it seems to be the case that the discriminant of $K$ is $p^{[K:\mathbb{Q}]-1}$  (according to Sage).  How can I prove that/ is the proof implied by something written down somewhere? 
I have a feeling this is somewhat deep so I ask it here; if it's insufficiently deep I'm happy to close it and ask elsewhere.

REPLY [11 votes]: This would also work for non-abelian, and even non-Galois extensions: if $L/\mathbb{Q}$ is totally ramified and tamely ramified at $p$ and $K$ is an intermediate field, then $K$ is also totally tamely ramified at $p$, so the exponent of $p$ in the discriminant of $K$ is $[K:\mathbb{Q}]-1$ (Serre, Corps Locaux, Proposition 13).