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For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of K2OE . For quadratic fields whose discriminant has arbitarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form Q[ √ p1 · · · pk], where the primes pi 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 K2OE is zero for such fields. In the course of proving the theorem, we will see how the conditions can be easily computed. Acknowledgements. I would like to thank J.S. Milne for his help in the writing of this paper and P.E. Conner and J. Hurrelbrink for their helpful suggestions.

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