The Birch–Tate conjecture is based on algebraic K-theory

Algebraic K-theory

In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

 proposed by both Bryan John Birch

Bryan John Birch

Bryan John Birch F.R.S. is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture....

 and John Tate

John Tate

John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...

. It relates the value of a Dedekind zeta function

Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...

 at s = −1 to the order of K₂ of the ring of integers

Ring of integers

In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

, for a number field F.

Progress on this conjecture has been made as a consequence of work on Iwasawa theory

Iwasawa theory

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...

, and in particular of the proofs given for the so-called "main conjecture of Iwasawa theory."

The problem remains unsolved as of 2011.

External links

• "Birch Tate conjecture" by Springer Online Reference Works