On the Hermitian structure of the square root of the inverse different. (Sur la structure hermitienne de la racine carrée de la codifférente.) (French) Zbl 0789.11062

Let \(K\) be a number field which is Galois over \(\mathbb{Q}\), of odd degree and let \(G\) be its Galois group. There is a unique fractional ideal of \(K\) which is unimodular for the quadratic form \(\text{Trace}_{K/ \mathbb{Q}} (x^ 2)\). This ideal is the square root of the inverse different, and is denoted \(A_ K\). In this paper, we describe an explicit representative of the \(\mathbb{Z}[G]\)-isometry class of \((A_ K,\text{Trace}_{K/ \mathbb{Q}}(x^ 2))\), which depends only on the wildly ramified prime numbers \(p\) having a ramification index in \(K\) different from \(p\).


11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11E39 Bilinear and Hermitian forms
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