The 2-Sylow subgroup of K2OF for number fields F

Let F be a quadratic number field. We give a criterion, via Hilbert symbols, for an element of order two in the tame kernel of F to be a fourth power in the tame kernel of F. The result can be applied to compute the 8-rank of the tame kernel of F and the Tate kernel of an imaginary quadratic number field. We list the 8-ranks of K2OF for all quadratic number fields whose discriminants have exactly two odd prime divisors. In the case when F is an imaginary quadratic number field with the 8-rank of K2OF=0, the Tate kernel of F is given too. An application of our method to the maximal real subfield of a cyclotomic field is discussed. Numerical examples, in particular the examples of quadratic number fields F with 4-rank of K2OF= 8-rank of K2OF=2 illustrate our results.