Algebraic integer

This article is about the ring of complex numbers integral over . For the general notion of algebraic integer, see Integrality.
Not to be confused with algebraic element or algebraic number.

In Algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers). The set of all algebraic integers, A, is closed under addition and multiplication and therefore is a commutative subring of the complex numbers. The ring A is the integral closure of regular integers in complex numbers.

The ring of integers of a number field K, denoted by OK, is the intersection of K and A: it can also be characterised as the maximal order of the field K. Each algebraic integer belongs to the ring of integers of some number field. A number x is an algebraic integer if and only if the ring is finitely generated as an abelian group, which is to say, as a -module.


The following are equivalent definitions of an algebraic integer. Let K be a number field (i.e., a finite extension of , the set of rational numbers), in other words, for some algebraic number by the primitive element theorem.

Algebraic integers are a special case of integral elements of a ring extension. In particular, an algebraic integer is an integral element of a finite extension .




See also


  1. Marcus, Daniel A. (1977), Number fields, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90279-1, chapter 2, p. 38 and exercise 41.
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