The minimum discriminant of sixth degree totally complex algebraic number fields

Abstract

Inspired by previous results of Kaur, Hunter, and Mayer we developed a new method of determining the minimum value of the discriminant in absolute value of totally complex sixth degree algebraic number fields. It is 9747 and is attained only by the algebraic number field Q( θ), where θ = (2ω 2 + 5ω) 1 3 − 1) (ω − 1)) is a root of the monic irreducible polynomial x 6 − 3 x 5 + 4 x 4 − 4 x 3 + 4 x 2 − 2 x + 1 with discriminant −9747 and ω is a primitive third root of unity.