Class number formula

The class number formula relates a number field’s class number to the residue of its Dedekind zeta function and arithmetic invariants including its regulator, roots of unity, and discriminant.

Number field: A finite extension of the rational numbers. The formula applies to this field and its arithmetic invariants.
Analytic class number formula: An explicit identity connecting the residue of a number field’s Dedekind zeta function with its class number and other invariants. This is the standard residue identity that gives the formula’s explicit form.
Ernst Eduard Kummer: A nineteenth-century German mathematician known for work on algebraic number theory and Fermat’s last theorem. His ideal-theoretic investigations helped establish the setting for class-number questions.
Class number problem: The study of how class numbers vary across families of number fields, including whether they remain bounded or attain specified values. The formula provides analytic tools for estimating and studying these class numbers.
Analytic class number formula for function fields: An identity relating a global function field’s zeta function to its divisor class number and other invariants. It parallels the number-field formula in a setting with different archimedean and unit factors.
Ideal class group: The finite group of fractional ideals modulo principal fractional ideals in a number field. Its order is the class number appearing in the formula.
Dedekind zeta residue: The coefficient of the simple pole at one in a number field’s Dedekind zeta function. The residue is the analytic side of the class number identity.
Richard Dedekind: A German mathematician who developed ideals and foundational concepts in algebraic number theory. His theory of ideals made class groups precise objects in the formula’s arithmetic side.
Minkowski bound: An explicit bound ensuring every ideal class in a number field contains an integral ideal of bounded norm. It offers a complementary route to computing class numbers that the formula can check.
Artin L-function: A complex L-function attached to a finite-dimensional representation of a Galois group. Artin L-functions refine zeta-factor relations beyond the single field identity.

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