The Birch–Tate conjecture is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

(more specifically in algebraic K-theory) proposed by both Bryan John Birch and

John Tate John Tate may refer to: * John Tate (mathematician) (1925–2019), American mathematician * John Torrence Tate Sr. (1889–1950), American physicist * John Tate (Australian politician) (1895–1977) * John Tate (actor) (1915–1979), Australian act ...

Statement

In algebraic K-theory, the group ''K''₂ is defined as the center of the Steinberg group of the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

of a number field ''F''. ''K''₂ is also known as the tame kernel of ''F''. The Birch–Tate conjecture relates the

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

of this group (its number of elements) to the value of the Dedekind zeta function

\zeta_F

. More specifically, let ''F'' be a totally real number field and let ''N'' be the largest natural number such that the

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...

of ''F'' by the ''N''th

root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...

has an

elementary abelian In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...

2-group In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of ''n''-groups. In some of the literature, 2-groups are also called gr-categories or groupal ...

as its Galois group. Then the conjecture states that :

\#K_2 = , N\zeta_F(-1), .

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."

References

* J. T. Tate, ''Symbols in Arithmetic'', Actes, Congrès Intern. Math., Nice, 1970, Tome 1, Gauthier–Villars(1971), 201–211

External links

* {{DEFAULTSORT:Birch-Tate conjecture Conjectures K-theory Unsolved problems in mathematics