Galois stability for finite groups
by
D. A. Malinin
Belorussian State Pedag. University, Minsk

For a given field of characteristic 0 we consider Galois extension E/F of finite degree d and finite abelian subgroups G subset GL_n(E) of the given exponent such that G is stable under the natural action of the Galois group of E/F. Let F(G) be a field obtained by adjoining to F all matrix coefficients of all g in G.

Theorem 1. Let \zeta_t be a primitive t-root of 1, let d > 1, t > 1, n >= [E(\zeta_t):E]d be the given integers, and let E be the given normal extension of F having the Galois group \Gamma and degree d. Then there is an abelian \Gamma- stable subgroup G subset GL_n(E) of the exponent t such that E=F(G).

Let \Cal O_F and \Cal O_E denote the maximal orders of number fields F and E, F=Q or F=Q(\surdd), d < 0, d in Z. Set \Cal O(N)={\alpha in \Cal O_F| |N_F/Q(\alpha)| <= N}. Let v(N) denote the total number of polynomials of degree m with coefficients in \Cal O(N), and let \psi(N) denote the number of those polynomials whose splitting fields do not contain any fields F(G) =/= F for G subset GL_n(\Cal O_E), and fixed n.

Theorem 2.

lim N --> \infty  (\psi(N)/v(N))=1.

Date received: April 20, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-48.