The Birch-Tate conjecture, in algebraic K-theory, is a conjecture proposed by Bryan John Birch and John Tate relating the K2 group of a number field to its Dedekind zeta function. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor a totally real number field F, let N be the largest positive integer such that adjoining the Nth roots of unity to F produces an extension whose Galois group is an elementary abelian 2 group. The Birch-Tate conjecture states that the order of the K2 group of the ring of integers of F equals the absolute value of N times the Dedekind zeta function of F evaluated at negative 1. 1 Progress Toward ResolutionThe conjecture remains open in general, though progress has followed from work on Iwasawa theory, in particular from proofs of the main conjecture of Iwasawa theory. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
Sources
1. Birch-Tate Conjecture (Wikipedia)
Wikimedia FoundationLead section
the Birch-Tate conjecture is a conjecture proposed by Bryan John Birch and John Tate relating the
Statement section
let F be a totally real number field and let N be the largest positive integer such that the extension of F by the N-th root of unity has an elementary abelian 2-group as its Galois group
Status section
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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