1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of $K{₂}_E$. For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form $\mathbb{Q}\left(\surd (p₁...p_k)\right)$, where the primes $p_i$ are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of $K{₂}_E$ is zero for such fields. In the course of proving...