## 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 $$G$$.

### MSC:

 1.1e+82 Algebraic theory of quadratic forms; Witt groups and rings 110000 Quadratic forms over general fields

### Keywords:

quadratic forms; trace forms; Burnside ring
Full Text:

### References:

 [1] J. Alperin, B. Bell. : Groups and Representations. Graduate texts in mathematics. Springer, New York, 1995. · Zbl 0839.20001 [2] P. Beaulieu, T. Palfrey. : The Galois number. Math. Ann., 309:81-96, 1997. · Zbl 0885.11027 [3] P. E. Conner, R. Perlis. : A Survey of Trace Forms of Algebraic Number Fields. World Scientific, Singapore, 1984. · Zbl 0551.10017 [4] Christof Drees, Martin Epkenhans, and Martin Krüskemper. : On the computation of the trace form of some Galois extensions. J. Algebra, 192:209-234, 1997. · Zbl 0873.11027 [5] Martin Epkenhans, Martin Krüskemper. : On Trace Forms of étale Algebras and Field Extensions. Math. Z., 217:421-434, 1994. · Zbl 0821.11024 [6] D. Gorenstein. : Finite Groups. Harper and Row, New York, 1968. · Zbl 0367.20007 [7] D. W. Lewis. : Witt rings as integral rings. Invent. Math., 90:631-633, 1987. · Zbl 0629.10017 [8] Ernst Witt. : Konstruktion von galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordnung $$p^f$$. J. Reine Angew. Math., 174:237-245, 1936. · Zbl 0013.19601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.