# Algebraic integer

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In 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 is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A. 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 [x] is finitely generated as an abelian group, which is to say, as a -module.

## Definitions

The following are equivalent definitions of an algebraic integer. Let K be a number field (i.e., a finite extension of ${\displaystyle \mathbb {Q} }$, the set of rational numbers), in other words, ${\displaystyle K=\mathbb {Q} (\theta )}$ for some algebraic number ${\displaystyle \theta \in \mathbb {C} }$ 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 ${\displaystyle K/\mathbb {Q} }$.

## Examples

${\displaystyle {\begin{cases}1,\alpha ,{\frac {\alpha ^{2}\pm k^{2}\alpha +k^{2}}{3k}}&m\equiv \pm 1\mod 9\\1,\alpha ,{\frac {\alpha ^{2}}{k}}&\mathrm {else} \end{cases}}}$

## Non-example

• If P(x) is a primitive polynomial which has integer coefficients but is not monic, and P is irreducible over Q, then none of the roots of P are algebraic integers. (Here primitive is used in the sense that the highest common factor of the set of coefficients of P is 1; this is weaker than requiring the coefficients to be pairwise relatively prime.)

## Facts

• The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. The monic polynomial involved is generally of higher degree than those of the original algebraic integers, and can be found by taking resultants and factoring. For example, if x2 − x − 1 = 0, y3 − y − 1 = 0 and z = xy, then eliminating x and y from z − xy and the polynomials satisfied by x and y using the resultant gives z6 − 3z4 − 4z3 + z2 + z − 1, which is irreducible, and is the monic polynomial satisfied by the product. (To see that the xy is a root of the x-resultant of z − xy and x2 − x − 1, one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)
• Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible quintics are not. This is the Abel-Ruffini theorem.
• Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring which is integrally closed in any of its extensions.