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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 \( \mathbb Q \), the set of rational numbers), in other words, \( K = \mathbb{Q}(\theta) \) for some algebraic number \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. See quadratic integers for more.

  • The ring of integers of the field \( F = \mathbf Q[\alpha], \alpha = \sqrt[3] m has the following integral basis, writing m = hk^2 for two square-free coprime integers h and k:[1]

    \( \begin{cases} 1, \alpha, \frac{\alpha^2 \pm k^2 \alpha + k^2}{3k} & m \equiv \pm 1 \mod 9 \\ 1, \alpha, \frac{\alpha^2}k & \mathrm{else} \end{cases} \)

    If ζn is a primitive n-th root of unity, then the ring of integers of the cyclotomic field Q(ζn) is precisely Z[ζn].
    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

    See also

    Integral element
    Gaussian integer
    Eisenstein integer
    Root of unity
    Dirichlet's unit theorem
    Fundamental units

    References

    Marcus, Daniel A. (1977), Number fields, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90279-1, chapter 2, p. 38 and exercise 41.

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

    Mathematics Encyclopedia

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