Dictionary

algebraic number theory en · NOUN

Etymology

Blend of algebraic number + number theory. Possibly (alternatively or also) a calque of German algebraische Zahlentheorie.

Meanings

1. (uncountable) The branch of number theory in which number-theoretic questions are expressed in terms of properties of algebraic number fields or related objects, and studied using techniques from algebra.
   • As is well-known in algebraic number theory, there is a far-reaching analogy between number fields and algebraic function fields in one variable with finite constant field. 1998, Norbert Klingen, Arithmetical Similarities: Prime Decomposition and Finite Group Theory, Oxford University Press (Clarendon Press), page 241:
   • Elementary arithmetic studies divisibility and factorization of ordinary integers, Algebraic number theory considers the same questions for algebraic numbers, solutions to polynomial equations with integer coefficients. In this setting, (un)fortunately, the uniqueness of prime factorization no longer holds. Remedying and measuring its failure is the starting point of algebraic number theory. 2013, Mak Trifković, Algebraic Theory of Quadratic Numbers, Springer, page v:
   • The field of mathematics that these generalize to is known as algebraic number theory. As a matter of historical fact, our development here parallels the development of algebraic number theory. 2008, Steven H. Weintraub, Factorization: Unique and Otherwise, CRC Press, page 143:

Forms

Relateds

arithmetic geometry · diophantine geometry · analytic number theory

Translations (9)

pl algebraiczna teoria liczb (branch of number theory concerning algebraic number fields) · cmn 代數數論 /代数数论 (branch of number theory concerning algebraic number fields) · hy հանրահաշվական թվերի տեսություն (branch of number theory concerning algebraic number fields) · pt teoria algébrica dos números (branch of number theory concerning algebraic number fields) · it teoria algebrica dei numeri (branch of number theory concerning algebraic number fields) · pt teoria dos números algébricos (branch of number theory concerning algebraic number fields) · fi algebrallinen lukuteoria (branch of number theory concerning algebraic number fields) · it teoria dei numeri algebrica (branch of number theory concerning algebraic number fields) · de algebraische Zahlentheorie (branch of number theory concerning algebraic number fields)