4-Ranks of K(2) of rings of integers in quadratic number fields.

Abstract

Let F be a quadratic extension of $\doubq$ and ${\cal O}\sb{F}$ the ring of integers in F. The group $K\sb2{\cal O}\sb{F}$ is finite and abelian. A result of Tate enables one to compute the number of elements of order two in $K\sb2{\cal O}\sb{F}$ by computing the number of elements of order two in the class group of F. Formulas for the number of elements of order four in $K\sb2{\cal O}\sb{F}$ are scarce. In this thesis, we prove a pair of conjectures posed by Conner and Hurrelbrink which, for certain classes of quadratic number fields, give explicit conditions under which $K\sb2{\cal O}\sb{F}$ has no elements of order four. Along the way we establish upper and lower bounds for the number of elements of order four in $K\sb2{\cal O}\sb{F}$ for any quadratic number field in terms of the number of elements of order four in the narrow class group. As there are formulas for the latter, our bounds are explicit.