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Algebraic integer

In number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in ℤ (the set of integers). The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A. The ring A is the integral closure of regular integers ℤ in complex numbers.

The ring of integers of a number field K, denoted by OK , is the intersection of K and A: it can also be characterised as the maximal order of the field K. Each algebraic integer belongs to the ring of integers of some number field. A number x is an algebraic integer if and only if the ring ℤ[x] is finitely generated as an abelian group, which is to say, as a ℤ-module.

Definitions

The following are equivalent definitions of an algebraic integer. Let K be a number field (i.e., a finite extension of \( \mathbb Q) \), in other words, \( K = \mathbb{Q}(\theta) \) for some \( \theta \in \mathbb{C} \) by the primitive element theorem.

\( \alpha \in K \) is an algebraic integer if there exists a monic polynomial \( f(x) \in \mathbb{Z}[x] \) such that f(\alpha) = 0.
\( \alpha \in K \) is an algebraic integer if the minimal monic polynomial of \( \alpha \) over \( \mathbb Q \) is in \mathbb{Z}[x].
\( \alpha \in K \) is an algebraic integer if \( \mathbb{Z}[\alpha] \) is a finitely generated \( \mathbb Z \)-module.
\( \alpha \in K \) is an algebraic integer if there exists a finitely generated \( \mathbb{Z} \) -submodule M \subset \mathbb{C} such that \alpha M \subseteq M.

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 \( K / \mathbb{Q}.\)

Examples

The only algebraic integers which are found in the set of rational numbers are the integers. In other words, the intersection of Q and A is exactly Z. The rational number a/b is not an algebraic integer unless b divides a. Note that the leading coefficient of the polynomial bx − a is the integer b. As another special case, the square root √n of a non-negative integer n is an algebraic integer, and so is irrational unless n is a perfect square.
If d is a square free integer then the extension K = Q(√d) is a quadratic field of rational numbers. The ring of algebraic integers OK contains √d since this is a root of the monic polynomial x² − d. Moreover, if d ≡ 1 (mod 4) the element (1 + √d)/2 is also an algebraic integer. It satisfies the polynomial x² − x + (1 − d)/4 where the constant term (1 − d)/4 is an integer. The full ring of integers is generated by √d or (1 + √d)/2 respectively.
If ζ_n is a primitive n-th root of unity, then the ring of integers of the cyclotomic field Q(ζn) is precisely Z[ζ_n].
If α is an algebraic integer then \( \beta=\sqrt[n]{\alpha} \) is another algebraic integer. A polynomial for β is obtained by substituting xn in the polynomial for α.

Non-example

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 primitive is used in the sense that the highest common factor of the set of coefficients of P 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 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. (To see that the xy is a root of the x-resultant of z − xy and x² − x − 1, one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)

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 A is a Bézout domain.

References

Daniel A. Marcus, Number Fields, third edition, Springer-Verlag, 1977