Birch–Tate conjecture

The Birch–Tate conjecture relates the order of the second algebraic K-group of a number field’s ring of integers to a special value of its Dedekind zeta function at zero.

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Algebraic K-theory: A field of mathematics that assigns groups and other invariants to rings and spaces. The conjecture’s group K₂ is an invariant defined within this broad framework.
Number field: A finite extension of the rational numbers. The conjecture is formulated separately for each number field.
John Tate: An American mathematician whose work shaped algebraic number theory and arithmetic geometry. Tate formulated the conjectural relation between K₂ and zeta values.
Mazur–Wiles theorem: A theorem proving the main conjecture of Iwasawa theory for the rational numbers. It established the conjectural formula for the rational field.
K₂ of a number ring: The second algebraic K-group of the ring of integers in a number field. Its finite order is the arithmetic quantity the conjecture predicts.
Ring of integers: The integral closure of the integers in a number field. This is the ring whose second K-group appears in the statement.
Bryan Birch: A British mathematician known for work in number theory, including the Birch–Swinnerton-Dyer conjecture. Birch’s calculations helped motivate the conjecture’s formulation.
Wiles's proof of Fermat's Last Theorem: Andrew Wiles’s proof, completed with Richard Taylor, of Fermat’s Last Theorem. The proof’s Iwasawa-theoretic methods also yielded cases of Birch–Tate.
Dedekind zeta function: A meromorphic function encoding the prime-ideal arithmetic of a number field. The conjecture extracts arithmetic information from its value at zero.
Finite abelian group: A finite group in which the group operation is commutative. For number rings, K₂ is finite, so its order is meaningful.