Algebraic integer
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In mathematics, an algebraic integer is a number which is a root of a polynomial with integer coefficients (that is, an algebraic number) with leading coefficient 1 (a monic polynomial). This generalizes the distinction between an integer n (the root of x - n = 0) and a fraction a/b (the root of bx - a = 0).
In more abstract terms, the ring of algebraic integers is the integral closure of the ring of integers in the field of algebraic numbers. A number x is an algebraic integer iff Z[x] is finitely generated as an abelian group, which is to say, Z-module.
Algebraic integers always belong to the ring of integers of some algebraic number field, for example the Gaussian integers lie in the field of Gaussian rationals and Eisenstein integers in the field Q(√−3).
[edit] Closure
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 the word primitive means that coefficients of P are coprime, so you can't simply divide out some integer constant. (Primitivity means that the greatest common divisor of the set of coefficients of P is 1; this is weaker than requiring the coefficients to be pairwise relatively prime.)
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 irredicible, and is the monic polynomial satisfied by the product.
An integer root of an algebraic integer is also an algebraic integer; the polynomial for such a root may easily be obtained by substiting x=yn in the polynomial for x and factoring. For example, substituting x=y2 into x2+x+1 and factoring leads to (y2+y+1)(y2-y+1), which are the irreducible polynomials for the various square roots of the roots x.
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: most roots of irreducible quintics are not.
More generally, 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 extension.
The algebraic integers are a Bézout domain.
Richard Schroeppel has demonstrated that if a number is constructible from the integers with roots, addition, and multiplication and division, and it is still an algebraic integer, then it is constructible without division. For example, the golden ratio, φ, is
![\frac{1 + \sqrt{5}}{2} = \sqrt[3]{2 + \sqrt{5}} = \sqrt[3]{1 + 2 \varphi}](../../../math/7/5/7/757cfd60e7953994f3523f90e3edbd70.png)
.
[edit] Reference
For the theorem by Schroeppel: Eric W. Weisstein, Radical Integer at MathWorld. The claim in Weisstein about cubics is mistaken; and radical integer is a nonce word.
