The totally real {$A\sb 5$} extension of degree {$6$} with minimum discriminant

Abstract

We determine the totally real algebraic number field $F$ of degree 6 with Galois group $A_5$ and minimum discriminant, showing that it is unique up to isomorphism and that it is generated by a root of the polynomial $$ f(t) = t^6 - 10 t^4 + 7 t^3 + 15 t^2 - 14 t + 3 $$ over the rationals. We also list the fundamental units and class number of $F$, as well as data for several other fields that arose in our computations and that might be of interest.