Dedekind zeta function

The Dedekind zeta function of a number field is a complex function whose Dirichlet series sums reciprocal powers of the norms of its nonzero integral ideals.

Ideal norm: The size of the quotient of a number ring by a nonzero integral ideal. Each ideal contributes a power of this integer to the defining Dirichlet series.
Number field: A finite-degree field extension of the rational numbers. Each number field has its own Dedekind zeta function.
Dedekind zeta function at zero: The value at zero of a number field's Dedekind zeta function, interpreted through its analytic continuation. The analytic class number formula expresses this value using the field's class number and unit data.
Riemann zeta function: The function defined for real part of s greater than one by the sum of n⁻ˢ over positive integers. It is the Dedekind zeta function of the rational number field.
Euler product: A factorization of a Dirichlet series into products indexed by prime objects. Unique factorization of ideals turns the zeta series into a product over prime ideals.
Ring of integers: The subring of a number field consisting of elements integral over the integers. The function sums ideals in this ring rather than ideals in the field itself.
Analytic class number formula: A formula relating the residue of a number field's Dedekind zeta function at one to arithmetic invariants. Its residue reveals the class number, regulator, roots of unity, and discriminant.
Dirichlet L-function: A Dirichlet series weighted by a periodic character on the integers. Its character weights distinguish residue classes, unlike the unweighted ideal count.
Prime ideal: A proper ideal whose quotient ring is an integral domain. Prime ideals supply the local factors in the function's Euler product.
Fractional ideal: A nonzero finitely generated module over a domain that becomes an integral ideal after multiplication by a nonzero element. Fractional ideals form the group whose classes measure failures of unique factorization.

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