On the relative discriminant of a cyclic extension of an algebraic number field whose relative degree is a prime number Let $K$ be a cyclic extension of an algebraic number field $k$ whose relative degree is a prime number $l$.
Hasse wrote(see below) in his "Bericht" that the relative discriminant of $K/k$ is of the form $\mathfrak f^{l-1}$, where $\mathfrak f$ is an integral ideal of $k$.
How do you prove this?
Bericht, p.22

Es sei also $k$ ein beliebiger Grundkörper und $K$ ein relativ-zyklischer
  Körper vom Primzahlgrade $l$ über $k$. Man kann zunächst
  zeigen, daß die Relativdiskriminante $\mathfrak d$ von $K$ nach $k$ die Form $\mathfrak d = \mathfrak f^{l-1}$ hat, wo $\frak f$ ein ganzes Ideal aus $k$ ist.

By the way, I'm not conversant with German. I just decoded this using Google.
 A: This follows from Hilbert's Different Formula. If $P'|P$ are primes of $K|k$, then the different exponent $d(P'|P)$ is
$$
d(P'|P)=\sum_{i\ge0}\left(\operatorname{ord} G_i(P'|P)-1\right),
$$
where $G_i(P'|P)$ are the higher ramification groups. Those are always subgroups of the Galois group $G=C_\ell$. So each and every term on the r.h.s. is either $\ell-1$ or $0$. Therefore all the different exponents will be multiples of $\ell-1$.
The relative discriminant is the norm of the different of $K/k$, so after taking the norm, we get an ideal of $k$ raised to power $\ell-1$.
A: Another way to see this (probably a bit overpowered though) is to use the conductor-discriminant formula: the discriminant of an abelian extension is the product of the conductors of all the characters of the Galois group.  In this case, when the Galois group is cyclic of prime order $l$, there are $l - 1$ non-trivial characters which are all have the same splitting field, namely $K$ itself, and so all have the same conductor, say $\mathfrak f$; and there is the trivial character, which has trivial conductor. So the discriminant is a product of $l-1$ copies of the same ideal $\mathfrak f$.
(From the point of view of building up the theory from first principles, and not using sledge hammers to smash nuts, the conductor-discriminant formula is not always the best way to prove things, since its proof comes fairly late in the theory of abelian extensions.  But from the point of view of actual computations, or reality checking statements, it can be very useful.)
