Algebraic Number Theory

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16 domain-specific abstractions whose origin domain is Algebraic Number Theory.

• Algebraic number field — A finite-degree field extension of the rational numbers, carrying arithmetic through its ring of integers, embeddings, ideals, norms, traces, units, and completions.
• Biquadratic field — A degree-four Galois extension of the rational numbers with Klein-four Galois group.
• Brauer's theorem on forms — A theorem guaranteeing large linear spaces of common zeros for sufficiently many-variable homogeneous forms over fields with bounded diagonal-form obstruction.
• Class number formula — A number-theoretic identity relating a Dedekind zeta function’s special behavior to a field’s class number, regulator, roots of unity, embeddings, and discriminant.
• Dedekind zeta function — Encode the nonzero ideals of a number field in a Dirichlet series and Euler product whose analytic behavior carries arithmetic information about the field.
• Different ideal — An ideal measuring the failure of the ring of integers of a number field to be self-dual under the trace pairing, inverse to the codifferent fractional ideal.
• Eisenstein reciprocity — A higher-power reciprocity law relating residue symbols in cyclotomic integer rings.
• Elliptic unit — A distinguished algebraic unit in an abelian extension of an imaginary quadratic field, constructed from special values of modular or elliptic functions and forming an Euler system.
• Fundamental unit (number theory) — A generator, modulo roots of unity, of the rank-one unit group of a number field's ring of integers.
• Golden field — The real quadratic number field obtained by adjoining the square root of five to the rationals.
• Heegner number — One of the nine square-free positive integers d for which the imaginary quadratic field Q(√−d) has class number one, equivalently unique factorization in its ring of integers.
• Local class field theory — The theory that classifies finite abelian extensions of a local field through a reciprocity map from the field's multiplicative group to its abelian Galois group.
• Local field — A nondiscrete locally compact Hausdorff topological field, equivalently in the non-Archimedean case a complete discretely valued field with finite residue field, serving as a completion-scale model of global arithmetic.
• Mahler measure — A multiplicative height-like measure of a polynomial equal to its leading coefficient magnitude times the moduli of roots outside the unit circle.
• Modulus (algebraic number theory) — A finite formal product of places of a global field encoding congruence and ramification conditions for ray class groups and abelian extensions.
• Reciprocity law — A number-theoretic rule relating splitting behavior of primes in an algebraic extension to congruence or residue information, generalizing quadratic reciprocity.