The totally real $A_6$ extension of degree 6 with minimum discriminant

Abstract

The totally real algebraic number field $F$ of degree 6 with Galois group $A_6$ and minimum discriminant is determined. It is unique up to isomorphy, and is generated by a root of the polynomial $t^6 - 24 t^4 + 21 t^2 + 9 t + 1$ over the rationals. We also give an integral basis and list the fundamental units and class number of $F$.