Algebraic integer

Definitions

• is an algebraic integer if there exists a monic polynomial such that.
• is an algebraic integer if the minimal monic polynomial of over is in.
• is an algebraic integer if is a finitely generated -module.
• is an algebraic integer if there exists a non-zero finitely generated -submodule such that.

Examples

• The only algebraic integers which are found in the set of rational numbers are the integers. In other words, the intersection of and is exactly. The rational number is not an algebraic integer unless divides. Note that the leading coefficient of the polynomial is the integer. As another special case, the square root of a nonnegative integer is an algebraic integer, but is irrational unless is a perfect square.
• If is a square-free integer then the extension is a quadratic field of rational numbers. The ring of algebraic integers contains since this is a root of the monic polynomial. Moreover, if, then the element is also an algebraic integer. It satisfies the polynomial where the constant term is an integer. The full ring of integers is generated by or respectively. See quadratic integers for more.
• The ring of integers of the field,, has the following integral basis, writing for two square-free coprime integers and :
• If is a primitive th root of unity, then the ring of integers of the cyclotomic field is precisely.
• If is an algebraic integer then is another algebraic integer. A polynomial for is obtained by substituting in the polynomial for.

  Non-example

• If is a primitive polynomial which has integer coefficients but is not monic, and is irreducible over, then none of the roots of are algebraic integers. Here primitive is used in the sense that the highest common factor of the set of coefficients of 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, and, then eliminating and from and the polynomials satisfied by and using the resultant gives, which is irreducible, and is the monic polynomial satisfied by the product.
• 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.
• The ring of algebraic integers is a Bézout domain, as a consequence of the principal ideal theorem.
• If the monic polynomial associated with an algebraic integer has constant term 1 or -1, then the reciprocal of that algebraic integer is also an algebraic integer, and is a unit, an element of the group of units of the ring of algebraic integers.