8-Ranks ofK2of Rings of Integers in Quadratic Number Fields Original Research Article

LetFbe a quadratic extension of image and imageFthe ring of integers inF. A result of Tate enables one to compute the 2-rank ofK2imageFin terms of the 2-rank of the class group. Formulas for the 4-rank ofK2imageFexist, but are more involved. We give upper and lower bounds on the 8-rank ofK2imageFin terms of the narrow class group. In certain cases the upper and lower bounds agree, and the 8-rank ofK2imageFis exactly the 8-rank of the narrow class group. We then give a family of fields for which this equality holds.