The Computation of Sextic Fields with a Cubic Subfield and no Quadratic Subfield

Abstract

We describe six tables of sixth-degree fields K containing a cubic subfield k and no quadratic subfield: one for totally real sextic fields, one for sextic fields with four real places, two for sextic fields with two real places, and two for totally imaginary sextic fields (depending on whether the cubic subfield is totally real or not). The tables provide for each possible discriminant dg of K a quadratic polynomial which defines K/k , the discriminant of the cubic subfield and the Galois group of a Galois closure AT/Q of K/Q. subfield. This is discussed in §4. In the second step, we remove the reducible polynomials and those which define a sextic field containing a quadratic subfield (these fields are in the tables

A Database for Field Extensions of the Rationals

“…[26,17,18,19] have finished the computation of minimal discriminants of all signatures and all primitive Galois groups of degree 6. [27,4] compute the minimal fields for imprimitive groups of degree 6. This yields enough information to determine the minimal fields for all groups and all conjugacy classes of that degree.…”

Artin L-functions of small conductor

“…Extensive responses to 2 came shortly thereafter, with papers often focusing on a single degree n = φ(e). Early work for quartics, quintics, sextics, and septics include respectively [BF89,For91,BFP93], [SPDyD94], [Poh82,BMO90,Oli91,Oli92,Oli90], and [Lét95]. Further results towards 2 in higher degrees are extractable from the websites associated to [JR14a], [KM01], and [LMF].…”