discriminant of subfield of $\mathbb{Q}(\zeta_p)$

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).

The Führerdiskriminantenproduktformel tells you that is it the product of conductors of characters, but all but the trivial character must have conductor $p$.