Algebraic number field

An algebraic number field is a finite field extension of the rational numbers. Its elements are algebraic numbers, each satisfying a nonzero polynomial with rational coefficients.

Field extension: A field extension is a larger field containing a given field as a subfield. A number field is specifically a finite extension of the rational field.
Ring of integers: The ring of integers of a number field consists of its elements integral over the ordinary integers. It provides the natural domain for integer-like arithmetic inside the field.
Dedekind zeta function: The Dedekind zeta function of a number field encodes counts of its ideals in a Dirichlet series. Its analytic behavior reflects the field's primes, class group, and units.
Function field: A function field is a finite extension of a rational function field over a specified constant field. Function fields parallel number fields, replacing rational numbers with rational functions.
Algebraic number: An algebraic number is a complex number that satisfies a nonzero polynomial with rational coefficients. Every element of a number field is algebraic over the rationals.
Integral element: An element is integral over a ring if it satisfies a monic polynomial with coefficients in that ring. Integral elements define which field elements belong to its ring of integers.
Dirichlet unit theorem: Dirichlet's unit theorem describes the units of a number field's ring of integers as a finite group times a free abelian group. It gives the structure and rank of the field's invertible algebraic integers.
Local field: A local field is a field complete with respect to a discrete valuation and having a finite residue field, or a related archimedean structure. Completions of number fields at primes produce local fields for place-by-place analysis.
Minimal polynomial: The minimal polynomial of an algebraic element is its monic irreducible polynomial over the base field. It determines the degree and algebraic relations of a field generator.
Ideal: An ideal is an additive subgroup of a ring closed under multiplication by every ring element. Ideals allow factorization to remain useful when elements do not factor uniquely.

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