The Galois number.

*(English)*Zbl 0885.11027Let \(G\) 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 \(\kappa_G\).

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 \(\kappa_G\).

Reviewer: Detlev Hoffmann (BesanĂ§on)

##### 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 |