Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.
The ring of integers of a number field, denoted by, is the intersection of and : it can also be characterized as the maximal order of the field. Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring 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 be a number field, in other words, for some algebraic number by the primitive element theorem.

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.

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.
Note that if is a primitive polynomial that 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 coefficients of is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.

Examples

The only algebraic integers that 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. The leading coefficient of the polynomial is the integer.
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 integer 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.
Finite generation of ring extension

For any, the ring extension of the integers by, denoted by, is finitely generated if and only if is an algebraic integer.
The proof is analogous to that of the corresponding fact regarding algebraic numbers, with there replaced by here, and the notion of field extension degree replaced by finite generation ; the only required change is that only non-negative powers of are involved in the proof.
The analogy is possible because both algebraic integers and algebraic numbers are defined as roots of monic polynomials over either or, respectively.

Ring

The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. Thus the algebraic integers form a ring.
This can be shown analogously to the corresponding proof for algebraic numbers, using the integers instead of the rationals.
One may also construct explicitly the monic polynomial involved, which is generally of higher degree than those of the original algebraic integers, 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 equation satisfied by the product.

Integral closure

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 that is integrally closed in any of its extensions.
Again, the proof is analogous to the corresponding proof for algebraic numbers being algebraically closed.

Additional facts

Any number constructible out of the integers with roots, addition, and multiplication is 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.
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 each is a unit, an element of the group of units of the ring of algebraic integers.
If is an algebraic number then is an algebraic integer, where satisfies a polynomial with integer coefficients and where is the highest-degree term of. The value is an algebraic integer because it is a root of, where is a monic polynomial with integer coefficients.
If is an algebraic number then it can be written as the ratio of an algebraic integer to a non-zero algebraic integer. In fact, the denominator can always be chosen to be a positive integer. The ratio is, where satisfies a polynomial with integer coefficients and where is the highest-degree term of.
The only rational algebraic integers are the integers. That is, if is an algebraic integer and then. This is a direct result of the rational root theorem for the case of a monic polynomial.