On the Simplest Quartic Fields and Related Thue Equations

Abstract

Let K be a field of char K = 2. For a ∈ K, we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial X 4 −aX 3 −6X 2 +aX +1 over K as the special case of the field intersection problem via multi-resolvent polynomials. From this result, over an infinite field K, we see that the polynomial gives the same splitting field over K for infinitely many values a of K. We also see by Siegel's theorem for curves of genus zero that only finitely many algebraic integers a ∈ OK in a number field K may give the same splitting field. By applying the result over the field Q of rational numbers, we establish a correspondence between primitive solutions to the parametric family of quartic Thue equationswhere m ∈ Z is a rational integer and c is a divisor of 4(m 2 + 16), and isomorphism classes of the simplest quartic fields.

A generalization of simplest number fields and their integral basis

“…E.Thomas [28], M.Mignotte [22], G.Lettl, A.Pethő and P.Voutier [21], [20] and G.Lettl and A.Pethő [19], I.Gaál [5] solved Thue equations corresponding to the simplest polynomials in absolute case, and C.Heuberger [11], I.Gaál, B.Jadrijević and L.Remete [6] in certain relative cases. A.Hoshi [12], [13], [14] gave a correspondence between solutions of a family of Thue equations and the isomorphism classes of the simplest number fields. He extended his results to a family of polynomials of degree 12, which has similar properties as the simplest polynomials, but over Q( √ −3).…”