On vanishing theorems for trace forms. (English) Zbl 1024.11021

The main result of this paper is a rank formula for the trace ideal of the Burnside ring of a finite group. Let \(B(G)\) be the Burnside ring of the finite group \(G\), and let \(N/K\) be a Galois extension of fields of characteristic different from two with Galois group isomorphic to \(G\). Then the trace forms of the subextensions of \(N/K\) give rise to a ring homomorphism from the Burnside ring \(B(G)\) into the Witt ring \(W(K)\). The intersection of the kernels of all such ring homomorphisms, when \(N/K\) runs over all Galois extensions with Galois group isomorphic to \(G\), is said to be the trace ideal \(T(G)\) of the ring \(B(G)\). Moreover, \(B(G)\) is a free Abelian group of finite rank, and it is proved that rank\((T(G))\) = rank\((B(G)) - c\), where \(c\) is the number of conjugacy classes of elements of order \(\leq 2\) in the group \(G\).


11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E04 Quadratic forms over general fields
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