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

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.

• The Führerdiskriminantenproduktformel tells you that is it the product of conductors of characters, but all but the trivial character must have conductor p. – Chris Wuthrich Oct 18 '19 at 12:31

The Führerdiskriminantenproduktformel tells you that is it the product of conductors of characters, but all but the trivial character must have conductor $$p$$.
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).
• @ElectricPenguin theres a very elementary definition of conductor in this special case: the smallest n s.t. $\chi$ factors through (Z/pZ)*-->(Z/nZ)* – dessin d'enfant terrible Oct 19 '19 at 22:52
• @dessind'efantterrible To split hairs, I would say that this is a computation of the conductor in this case rather than a definition. The distinction is that this if you actually want to prove the Führerdiskriminantenproduktformel (in the abelian case) for a local extension $L/K$, you really need to grapple with $N_{L/K}(L^*)$ in some way (say via basic local class field theory), whereas one could compute (with a full proof) the different of a totally tamely ramified extension $L/K$ in a single MO comment. – user145307 Oct 20 '19 at 0:33
• @GHfromMO The extension $\mathbf{Q}_p(p^{1/e})$ with $e > 1$ prime to $p$ (for example) is certainly totally tamely ramified and is never generated by a root of unity. Perhaps you are thinking about the fact that an abelian extension of $\mathbf{Q}_p$ is cyclotomic (local Kronecker-Weber)? – user145307 Oct 20 '19 at 0:44