# A proof of a theorem on the different in algebraic number fields

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 $D_{L/K} = \prod_{\sigma ∈ G - \{1\}}J_\sigma$.

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Of course it is a matter of taste, but one of the charms, to me, of mathematics is the surprise that new ideas and methods sometimes usefully bear upon pre-existing questions... that made no mention of the "new" ideas. For example, the Iwasawa-Tate treatment of "abelian" zeta and L-functions is much clearer, in the end, than Hecke's classical treatment (although Hecke's papers were very well written!) In particular, "Fujisaki's Compactness Lemma" gives the clearest proof of Dirichlet Units and finiteness of class numbers. Elementariness is not always simplicity. – paul garrett May 23 '12 at 0:22
From a reverse-mathematical perspective, knowing the result doesn't rely on p-adic completions is a great help! – David Roberts May 23 '12 at 1:00
David, note that there is already a proof not based on $p$-adic completions (using "Kronecker's method on forms"). I believe the OP wants a simpler, but still classical, proof. – B R May 23 '12 at 3:52
I agree with Paul Garrett; a proof using $p$-adic completions is likely to be conceptually simpler and more edifying that a proof artificially phrased in the language of ideals. So you should be happy with your $p$-adic proof. – Chandan Singh Dalawat May 23 '12 at 4:29
In the analogy of Paul Garrett, since I only know the Iwasawa-Tate treatment of L-functions, I'd like to know the Hecke's classical treatment. – Makoto Kato May 23 '12 at 17:48