On trace forms and the Burnside ring (Q2704197)

Let \(K\) be a field of characteristic not \(2\), let \(L\) be a finite separable extension of \(K\), and let \(N\) be the Galois closure of \(L/K\). Let \(G\) be the Galois group of \(N/K\) and \(H\) be the Galois group of \(N/L\). This paper is concerned with finding a monic integer polynomial \(p(X)\) of minimal degree, depending only on the Galois action of \(G\) of the set of left cosets \(G/H\), such that \(p(X)\) annihilates the trace form of \(L/K\) in the Witt ring \(W(K)\) of \(K\). The paper begins by discussing examples of annihilating trace forms in the existing literature, and then improves existing results by reducing to \(2\)-groups via Springer's theorem on the lifting of quadratic forms to odd degree extensions. In certain cases additional improvements are obtained by using explicit calculations of the trace ideal of the Burnside ring of \(G\). Some explicit examples are given.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00036].