Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers, which are roots of polynomials with rational coefficients. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or This article is about the zeros of a function which should not be confused with the value at zero. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions An algebraic number field is any finite (and therefore algebraic) field extension of the rational numbers. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. These fields contain elements analogous to the integers, the so-called algebraic integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This article deals with the ring of complex numbers integral over Z. In this setting, the familiar features of the integers (e. g. , unique factorization) need not hold. In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written The virtue of the machinery employed — Galois theory, group cohomology, class field theory, group representations and L-functions — is that it allows one to recover that order partly for this new class of numbers. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper In Mathematics, class field theory is a major branch of Algebraic number theory. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of The theory of L -functions has become a very substantial and still largely Conjectural, part of contemporary Number theory.

See also

• Arithmétique modulaire A survey of number theory, with applications (in French Wikipedia)
• List of algebraic number theory topics
• Algebraic number fields

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