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Galois module structure in weakly ramified 3-extensions. (English) Zbl 1075.11071

As B. Erez showed in 1991 [Math. Z. 208, No. 2, 239–255 (1991; Zbl 0713.11078)], the inverse different of an odd-degree Galois extension \(N/k\) is of the form \({\mathcal A}^2\), and the Galois module \(\mathcal A\) is locally free iff \(N/k\) is weakly ramified (which is to say: all second ramification groups are trivial). This is a pretty analog of Noether’s theorem about tame extensions (in which the role of \(\mathcal A\) is taken by \({\mathcal O}_N\)), and naturally leads to the question: When is the class of \(\mathcal A\) trivial in \(Cl({\mathcal O}_k[G])\) ? (As usual, \(G={}\)Gal\((N/k))\). Apparently no counterexample with \(k={\mathbb Q}\) has been found yet. In the present paper the author makes the extra assumption that \(N/{\mathbb Q}\) is a 3-extension, and he proves that then the order of the class of \(\mathcal A\) is 3 or 1.
The proof proceeds in the following steps: Via Fröhlich’s Hom description and references to earlier work, the problem is reduced to a \(p\)-adic problem (this part of the argument works for arbitrary odd primes \(p\) instead of 3). In particular, the Galois group is replaced by the inertia group \(\Gamma_0\) at \(p\). This makes the group abelian and the Hom description much simpler. One knows that a certain object \(k_p^p\), constructed via resolvents, is a unit of the maximal order of \({\mathcal O}_{N_0}[\Gamma_0]\), and the main result is then equivalent to the property that it is already in \({\mathcal O}_{N_0}[\Gamma_0]\). The latter property is then established for \(p=3\), using a simple integrality criterion (when does an element of the maximal order, expressed by its character values, already belong to the group ring?), and fairly long explicit calculations that involve, among other things, classical identities between the two standard types of symmetric functions (sums of products, and sums of powers).
It would be interesting to check by computer whether \(\mathcal A\) may have nontrivial class. In a previous publication [J. Number Theory 91, No.1, 126–152 (2001; Zbl 0997.11098)], the author already performed similar computations for certain extensions of degree 27.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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