α ∈ K is an algebraic integer if there exists a monic polynomial such that f(α) = 0.
α ∈ K is an algebraic integer if the
minimal monic polynomial of α over is in .
α ∈ K is an algebraic integer if is a finitely generated -module.
α ∈ K is an algebraic integer if there exists a non-zero finitely generated -
submodule such that αM ⊆ 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 .
Examples
The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of and A is exactly . The rational number a/b is not an algebraic integer unless bdividesa. The leading coefficient of the polynomial bx − a is the integer b. As another special case, the
square root of a nonnegative integer n is an algebraic integer, but is
irrational unless n is a
perfect square.
If d is a
square-free integer then the
extension is a
quadratic field of rational numbers. The ring of algebraic integers OK contains since this is a root of the monic polynomial x2 − d. Moreover, if d ≡ 1
mod 4, then the element is also an algebraic integer. It satisfies the polynomial x2 − x + 1/4(1 − d) where the
constant term1/4(1 − d) 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 , α = 3√m, has the following
integral basis, writing m = hk2 for two
square-freecoprime integers h and k:[1]
If α is an algebraic integer then β = n√α 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 that has integer coefficients but is not monic, and P is
irreducible over , then none of the roots of P are algebraic integers (but arealgebraic numbers). Here primitive is used in the sense that the
highest common factor of the 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 = 0 and the polynomials satisfied by x and y using the resultant gives z6 − 3z4 − 4z3 + z2 + z − 1 = 0, which is irreducible, and is the monic equation 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 that is
integrally closed in any of its extensions.
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.
If x is an algebraic number then anx is an algebraic integer, where x satisfies a polynomial p(x) with integer coefficients and where anxn is the highest-degree term of p(x). The value y = anx is an algebraic integer because it is a root of q(y) = an − 1 np(y /an), where q(y) is a monic polynomial with integer coefficients.
If x 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 |an|x / |an|, where x satisfies a polynomial p(x) with integer coefficients and where anxn is the highest-degree term of p(x).