Algebraic integer

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In number theory, an algebraic integer is a complex number which is a root of some monic (leading coefficient 1) polynomial with integer coefficients. The set of all algebraic integers is closed under ring operations so forms a subring of complex numbers denoted by A. The ring of algebraic integers (also called ring of integers) of a number field K is denoted by O_K and it's the intersection of K and A. Ring of integers is the integral closure of the ring of integers in the field of algebraic numbers. Each algebraic integer belongs to the ring of integers of some algebraic number field. A number x is an algebraic integer if and only if Z[x] is finitely generated as an abelian group, which is to say, Z-module.

1 Examples
2 Closure
3 Reference
4 See also

[edit] Examples

The only algebraic integers in rational numbers are the ordinary integers. In other words the intersection of Q and A is exactly Z. The rational number a/b is not an algeraic integer unless b is 1. Note that the leading coefficient of the polynomial bx - a =0 is the integer b.

If d is a square free integer then the extension K=Q(√d) is a quadratic field extension of rational numbers. The ring of algebraic integers O_K involves √d since it's a root of the monic polynomial x^2-d. Moreover in case d=1 (mod 4), the element (1+√d)/2 turns out out be an algebraic integer. It satisfies the polynomial x^2-x+(1-d)/4 where the constant term (1-d)/4 is an integer.

[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 irreducible, 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 substituting 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

$\frac{1 + \sqrt{5}}{2} = \sqrt[3]{2 + \sqrt{5}} = \sqrt[3]{1 + 2 \varphi}$ .