Classes of order 4 in the strict class group of number fields and remarks on unramified quadratic extensions of unit type

Abstract

Let $K$ be a number field of degree $n$ over ${\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\text{rank}_2 \, } ({ E_{K}^{+} } / E_K^2) - \lfloor n /2 \rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units and the totally positive units of $K$, respectively, and $\text{rank}_2$ is the dimension as an elementary abelian 2-group. In particular, the strict class group of a totally real field $K$ with a totally positive system of fundamental units contains at least$(n-1)/2$ ($n$ odd) or $n/2 -1$ ($n$ even) independent elements of order 4. We also investigate when units in $K$ are sums of two squares in $K$ or are squares mod 4 in $K$.