On the 2-primary part of K₂ of rings of integers in certain quadratic number fields

1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of ${\displaystyle K{₂}_E}$. 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 ${\displaystyle \mathbb{Q}(\surd (p₁...p_k))}$, where the primes ${\displaystyle p_i}$ 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 ${\displaystyle K{₂}_E}$ is zero for such fields. In the course of proving the theorem, we will see how the conditions can be easily computed.