I asked this question in Stack Exchange, here, but I got no answer so far. I don't know any modern book on algebraic number theory which states the following theorem, let alone its proof except Takagi's book written in Japanese. This is surprising to me, because I think the theorem is a fundamental fact in algebraic number theory. I know a proof of this theorem which uses a somewhat complicated argument using the Kronecker's method on forms. Moreover I think I came up with a proof using p-adic completions. So I'm looking for a proof using only ideals. Since this is a theorem concerning only ideals, it would be nice to have a proof without p-adic completions(I think ideals are more concrete than p-adic numbers).

**Theorem**
Let $K ⊂ L ⊂ E$ be a tower of algebraic number fields.
Suppose that $E/K$ is a Galois extension.
Let $B$ and $C$ be the rings of algebraic integers in $L$ and $E$ respectively.
Let $G$ be the Galois group of $E/K$.
Let $H$ be the Galois group of $E/L$.
Let $G/H$ be the set of left cosets of $G$ by $H$.
Let $S$ be the set of representatives of $G/H$ - {$H$}.
For each $\sigma ∈ S$, Let $J_\sigma$ be the ideal of $C$ generated by the set {$x - \sigma(x); x ∈ B$}.
Let $J$ be $\prod_{\sigma ∈ S}J_\sigma$.
Let $D_{L/K}$ be the different.
Then $(D_{L/K})C = J$.

As a corollary to this theorem, we get the following useful result.

**Corollary**
Let $L$ be a finite Galois extension of an algebraic number field K.
Let $B$ be the ring of algebraic integers in $L$.
Let $G$ be the Galois group of $L/K$.
For each $\sigma ∈ G$ - {1}, Let $J_\sigma$ be the ideal of $B$ genertated by the set {$x - \sigma(x); x ∈ B$}.
Let $D_{L/K}$ be the different.
Then $D_{L/K} = \prod_{\sigma ∈ G - \{1\}}J_\sigma$.

andfiniteness of class numbers. Elementariness is not always simplicity. – paul garrett May 23 '12 at 0:22