Abstract
LetFbe a quadratic extension of Q and OFthe ring of integers inF. A result of Tate enables one to compute the 2-rank ofK2OFin terms of the 2-rank of the class group. Formulas for the 4-rank ofK2OFexist, but are more involved. We give upper and lower bounds on the 8-rank ofK2OFin terms of the narrow class group. In certain cases the upper and lower bounds agree, and the 8-rank ofK2OFis exactly the 8-rank of the narrow class group. We then give a family of fields for which this equality holds.
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