Galois Groups of Even Sextic Polynomials

Let f (x) = x(6) + ax(4) + bx(2) + c be an irreducible sextic polynomial with coefficients from a field F of characteristic # 2, and let g(x) = x(3) + ax(2) + bx + c. We show how to identify the conjugacy class in S-6 of the Galois group off over F using only the discriminants of f and g and the reducibility of a related sextic polynomial. We demonstrate that our method is useful for producing one-parameter families of even sextic polynomials with a specified Galois group.