Algebraic integer
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In mathematics, an algebraic integer is a number which is the root of an integer polynomial (that is, an algebraic number) which also has the special property that the polynomial can be expressed with integer coefficients and 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. Examples of algebraic integers include the Gaussian integers and Eisenstein integers.
In algebraic terms, a number is an algebraic integer iff multiplication by it maps a complex set M which is a finitely generated module over the integers into itself.
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 of two algebraic integers is an algebraic integer, and their difference; so is their product. Their quotient usually isn't. (The monic polynomial involved is usually of higher degree than those of the original algrebraic integers.) An integer root of an algebraic integer is also an algebraic integer. Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer; but not all algebraic integers are so constructable: 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
- <math>\frac{1 + \sqrt{5}}{2} = \sqrt[3]{2 + \sqrt{5}} = \sqrt[3]{1 + 2 \phi} </math>.
Reference
For the theorem by Schroeppel: Eric W. Weisstein, Radical Integer at MathWorld. The claim in Weisstein about cubics is mistaken; and radical integeris a nonce word.