Birch–Tate conjecture

Statement

In algebraic K-theory, the group K₂ is defined as the center of the Steinberg group of the ring of integers of a number field F. K₂ is also known as the tame kernel of F. The Birch–Tate conjecture relates the order of this group (its number of elements) to the value of the Dedekind zeta function ζ F . More specifically, let F be a totally real number field and let N be the largest natural number such that the extension of F by the Nth root of unity has an elementary abelian 2-group as its Galois group. Then the conjecture states that

Status

Progress on this conjecture has been made as a consequence of work on Iwasawa theory, and in particular of the proofs given for the so-called "main conjecture of Iwasawa theory."

The problem remains unsolved as of 2011.