algebraic number theory

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

algebraic number theory (uncountable)

1. (mathematics, number theory) 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.
   • 1998, Norbert Klingen, Arithmetical Similarities: Prime Decomposition and Finite Group Theory‎^[1], Oxford University Press (Clarendon Press), page 241:

     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.

   • 2008, Steven H. Weintraub, Factorization: Unique and Otherwise, CRC Press, page 143:

     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.

   • 2013, Mak Trifković, Algebraic Theory of Quadratic Numbers, Springer, page v:

     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.

• analytic number theory
• arithmetic geometry
• diophantine geometry

•   Algebraic number field on Wikipedia.Wikipedia
•   Number theory on Wikipedia.Wikipedia
•   Ring of integers on Wikipedia.Wikipedia
• Algebraic number theory on Encyclopedia of Mathematics
• Algebraic Number Theory on Wolfram MathWorld