On the trace form of Galois algebras

We study the trace form qL of G‐Galois algebras L/K when G is a finite group and K is a field of characteristic different from 2. We introduce in this paper the category of 2‐reduced groups and, when G is such a group, we use a formula of Serre to compute the second Hasse–Witt invariant of qL. By combining this computation with work of Quillen we determine the isometry class of qL for large families of G‐Galois algebras over global fields. We also indicate how our results generalize to Galois G‐covers of schemes.