## The Galois number.(English)Zbl 0885.11027

Let $$G$$ be a finite group acting on a nonempty finite set $$X$$. The Galois number $$t_G$$ of this action is defined to be $$1+M$$, where $$M$$ denotes the maximum among all numbers which occur as the number of fixed points of those elements in $$G$$ which do not act as the identity. If $$f$$ is an irreducible separable polynomial over a field $$K$$ of degree $$n$$ and $$N$$ is the normal closure of $$K(\theta )$$ where $$\theta$$ is a root of $$f$$, then $$G=\text{Gal}(N/K)$$ acts on the roots of $$f$$ and the Galois number of this action is just the smallest $$j$$ such that any $$j$$ roots of $$f$$ generate $$N$$ over $$K$$ (cf. Corollary 2 in the present paper).
The authors study certain identities in the Burnside ring $$B(G)$$ that involve the above defined Galois numbers. In the situation $$G=\text{Gal}(N/K)$$, where $$\text{char}(K)\neq 2$$ and $$N$$ is the normal closure of a degree $$n$$ extension $$L/K$$, they use these identities and a homomorphism from $$B(G)$$ into the Witt ring $$W(K)$$ to obtain vanishing theorems for the trace form $$\langle L\rangle$$ of the extension $$L/K$$. (The construction of this homomorphism is explained in detail.) For instance, they obtain the Witt ring identity $\prod_{m=0}^{t-1}(\langle L\rangle -m\cdot\langle 1\rangle )= \kappa_G\cdot\langle N\rangle,$ where $$t=t_G$$ as above and $$\kappa_G= n!/((n-t)! [N:K])$$ (Theorem 2), from which they deduce that the polynomial $$(X-n\cdot 1)\prod (X-m\cdot 1)$$, $$m$$ ranging over the integers between $$0$$ and $$t-1$$ with $$m\equiv n\bmod 2$$, vanishes on $$\langle L\rangle$$. Except for the case $$t=n-1$$, this polynomial is of lower degree than other general annihilating polynomials of trace forms of degree $$n$$ that appeared previously in the literature [see J. Hurrelbrink, Can. Math. Bull. 32, No. 4, 412-416 (1989; Zbl 0636.10019)].
The authors conclude with a list of examples where they evaluate $$t_G$$ and $$\kappa_G$$ for certain finite groups and certain actions, and where they explicitly compute certain trace form identities under additional assumptions on $$K$$, $$L$$, $$t_G$$ and $$\kappa_G$$.

### MSC:

 11E04 Quadratic forms over general fields 19A22 Frobenius induction, Burnside and representation rings 20C15 Ordinary representations and characters 11E81 Algebraic theory of quadratic forms; Witt groups and rings

Zbl 0636.10019
Full Text: