In mathematics, specifically in algebraic K-theory, the Birch–Tate conjecture is a conjecture proposed by Bryan John Birch and John Tate relating the group of a field to its Dedekind zeta function.
Statement Formally, the group of a number field is defined as the center of the Steinberg group of the ring of integers of . is also known as the tame kernel of . The Birch–Tate conjecture relates the order of this group (its number of elements) to the value of the Dedekind zeta function .
More specifically, let be a totally real number field and let be the largest positive integer such that the extension of by the -th 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."
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