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For a relative-quadratic extension E/F of number fields, the 4-ranks of the tame kernel K2(OE) and of the positive tame kernel K2(OE) = ker(K2(OE) → ⊕w real μ2) of E are computed in terms of local Hilbert symbols provided that the positive tame kernel of the base field F has no 2-torsion. The method generalizes and improves results of Qin for quadratic number fields. Various applications include the determination of the Tate kernel, provided the 2-primary part of K2(OE) is elementary abelian, and partial results about the 8-rank of the tame kernel K2(OE) if E is not totally complex *.

Tame kernels under relative quadratic extensions and Hilbert symbols