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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 ramified and tamely ramified at $p$, so the exponent of $p$ in its discriminant is $[K:\mathbb{Q}]-1$ (Serre, Corps Locaux, Proposition 13).