turkmath.org

Let K be a field of characteristic different from 2, let G be a finite group and let L/K be a G-Galois extension. We attach to this extension the so called trace form. This is the G-quadratic form qL : L → K defined by qL(x) = Tr_{L/K}(x^2). When the degree of L/K is odd, Bayer and Lenstra have proved that L has a normal and self-dual basis over K; therefore qL is isometric to the unit form < 1,··· ,1 >. Their result does not generalize to extensions of even degree. This is the case we want to consider. In this talk, we recall the definition of the Hasse-Witt invariants of a quadratic form and we introduce the notion of 2-reduced groups. Using a formula of Serre and theorems of Quillen, we compute the Hasse-Witt invariants of qL when the Galois group of L/K is 2-reduced. As a consequence, we obtain infinite families of Galois extensions L/K, having a trace form qL isomorphic to the unit form. These results are part of a joint work with T. Chinburg, B. Morin and M.J. Taylor.