Galois groups of doubly even octic polynomials

Abstract

Let [Formula: see text] be an irreducible polynomial with rational coefficients, [Formula: see text] the number field defined by [Formula: see text], and [Formula: see text] the Galois group of [Formula: see text]. Let [Formula: see text], and let [Formula: see text] be the Galois group of [Formula: see text]. We investigate the extent to which knowledge of the conjugacy class of [Formula: see text] in [Formula: see text] determines the conjugacy class of [Formula: see text] in [Formula: see text]. We show that, in general, knowledge of [Formula: see text] does not automatically determine [Formula: see text], except when [Formula: see text] is isomorphic to [Formula: see text] (the cyclic group of order 4). In this case, we show [Formula: see text] is isomorphic to a non-split extension of [Formula: see text] (the dihedral group of order 8) by [Formula: see text]. We also show that [Formula: see text] is completely determined when [Formula: see text] is isomorphic to [Formula: see text] and [Formula: see text] is a perfect square. In this case, [Formula: see text].

Galois Groups of Reciprocal Sextic Polynomials

“…For example, Galois groups of even quartic polynomials (x 4 + ax 2 + b), even sextic polynomials (x 6 + ax 4 + bx 2 + c) and doubly even octic polynomials (x 8 + ax 4 + b) have elementary characterisations (see for example [1,2]). In each case, the characterisation leverages information about the index-2 subfield of the field defined by the polynomial.…”