On the Hermitian structure of the square root of the inverse different (Q1313316)

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 \(p\).