Remarks on \(K_ 2\) of number fields. (English) Zbl 0589.12010

Gras, Georges

J. Number Theory 23, 322-335 (1986).

Let k be a number field with ring of integers R and let \(H^ 0_ 2(k)\) be the kernel of the homomorphism \(K_ 2(R)\to \oplus_{v\quad real}\mu_ 2\) given by the Hilbert symbols at the real places. The author studies the triviality of the p-part of \(H^ 0_ 2(k)\) for Galois p-extensions of number fields. In particular for \(p=2\) or 3 a complete list of abelian p-extensions of \({\mathbb{Q}}\) with trivial p-part of \(H^ 0_ 2(k)\) is given.

Reviewer: M.Kolster

MSC:

11R70	\(K\)-theory of global fields
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

\(K_ 2\); Birch-Tate conjecture; tame kernel; Hilbert symbols; Galois p- extensions; number fields

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