The Galois group of a polynomial with two indeterminate coefficients

Stephen D. Cohen

Vol. 90 (1980), No. 1, 63–76

DOI: 10.2140/pjm.1980.90.63

Abstract

Suppose that f(x) = ∑ _i=0ⁿα_iXⁱ(α₀α_n≠0) is a polynomial in which two of the coefficients are indeterminates t, u and the remainder belong to a field F. We find the galois group of f over F(t,u). In particular, it is the full symmetric group S_n provided that (as is obviously necessary) f(X)≠f₁(X^r) for any r > 1. The results are always valid if F has characteristic zero and hold under mild conditions involving the characteristic of F otherwise. Work of Uchida [10] and Smith [9] is extended even in the case of trinomials Xⁿ + tX^a + u on which they concentrated.

Mathematical Subject Classification 2000

Primary: 12E05

Secondary: 12F10

Milestones

Received: 9 May 1979

Published: 1 September 1980