Algebraic integer

Template:Short description Template:Use mdy dates Template:Use American English 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 (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers Template:Mvar 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 Template:Mvar, denoted by Template:Mathcal_K, is the intersection of Template:Mvar and Template:Mvar: it can also be characterized as the maximal order of the field Template:Mvar. Each algebraic integer belongs to the ring of integers of some number field. A number Template:Mvar is an algebraic integer if and only if the ring $\mathbb{\mathbb{Z}}[\alpha ]$ is finitely generated as an abelian group, which is to say, as a $\mathbb{\mathbb{Z}}$-module.

Definitions

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

• α ∈ K is an algebraic integer if there exists a monic polynomial $f(x)\in \mathbb{\mathbb{Z}}[x]$ such that f(α) = 0.
• α ∈ K is an algebraic integer if the minimal monic polynomial of Template:Mvar over $\mathbb{\mathbb{Q}}$ is in $\mathbb{\mathbb{Z}}[x]$.
• α ∈ K is an algebraic integer if $\mathbb{\mathbb{Z}}[\alpha ]$ is a finitely generated $\mathbb{\mathbb{Z}}$-module.
• α ∈ K is an algebraic integer if there exists a non-zero finitely generated $\mathbb{\mathbb{Z}}$-submodule $M\subset \mathbb{ℂ}$ 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 $K/\mathbb{\mathbb{Q}}$.

Note that if P(x) is a primitive polynomial that has integer coefficients but is not monic, and Template:Mvar is irreducible over $\mathbb{\mathbb{Q}}$, then none of the roots of Template:Mvar are algebraic integers (but are algebraic numbers). Here primitive is used in the sense that the highest common factor of the coefficients of Template:Mvar 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 $\mathbb{\mathbb{Q}}$ and Template:Mvar is exactly $\mathbb{\mathbb{Z}}$. The rational number Template:Sfrac is not an algebraic integer unless Template:Mvar divides Template:Mvar. The leading coefficient of the polynomial bx − a is the integer Template:Mvar.
• The square root $\sqrt{n}$ of a nonnegative integer Template:Mvar is an algebraic integer, but is irrational unless Template:Mvar is a perfect square.
• If Template:Mvar is a square-free integer then the extension $K=\mathbb{\mathbb{Q}}(\sqrt{d})$ is a quadratic field of rational numbers. The ring of algebraic integers Template:Mathcal_K contains $\sqrt{d}$ since this is a root of the monic polynomial x² − d. Moreover, if d ≡ 1 mod 4, then the element $\frac{1}{2}(1+\sqrt{d})$ is also an algebraic integer. It satisfies the polynomial x² − x + Template:Sfrac(1 − d) where the constant term Template:Sfrac(1 − d) is an integer. The full ring of integers is generated by $\sqrt{d}$ or $\frac{1}{2}(1+\sqrt{d})$ respectively. See Quadratic integer for more.
• The ring of integers of the field $F=\mathbb{\mathbb{Q}}[\alpha ]$, α = Template:Radic, has the following integral basis, writing m = hk² for two square-free coprime integers Template:Mvar and Template:Mvar: $$\{\begin{array}{ll}1,\alpha ,\frac{{\alpha }^2\pm k^2\alpha +k^2}{3k}& m\equiv \pm 1mod9\\ 1,\alpha ,\frac{{\alpha }^2}{k}& \text{otherwise}\end{array}$$
• If Template:Mvar is a primitive Template:Mvarth root of unity, then the ring of integers of the cyclotomic field $\mathbb{\mathbb{Q}}({\zeta }_n)$ is precisely $\mathbb{\mathbb{Z}}[{\zeta }_n]$.
• If Template:Mvar is an algebraic integer then β = Template:Radic is another algebraic integer. A polynomial for Template:Mvar is obtained by substituting xⁿ in the polynomial for Template:Mvar.

Finite generation of ring extension

For any α, the ring extension (in the sense that is equivalent to field extension) of the integers by α, denoted by $\mathbb{\mathbb{Z}}[\alpha ]\equiv \{\sum _{i=0}^n{\alpha }^i{z}_i|z_i\in \mathbb{\mathbb{Z}},n\in \mathbb{\mathbb{Z}}\}$, is finitely generated if and only if α is an algebraic integer.

The proof is analogous to that of the corresponding fact regarding algebraic numbers, with $\mathbb{\mathbb{Q}}$ there replaced by $\mathbb{\mathbb{Z}}$ here, and the notion of field extension degree replaced by finite generation (using the fact that $\mathbb{\mathbb{Z}}$ is finitely generated itself); the only required change is that only non-negative powers of α 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 $\mathbb{\mathbb{Z}}$ or $\mathbb{\mathbb{Q}}$, 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 $\mathbb{\mathbb{Z}}$ instead of the rationals $\mathbb{\mathbb{Q}}$.

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 x² − x − 1 = 0, y³ − y − 1 = 0 and z = xy, then eliminating Template:Mvar and Template:Mvar from z − xy = 0 and the polynomials satisfied by Template:Mvar and Template:Mvar 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 Template:Mvar is a root of the Template:Mvar-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.)

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 x is an algebraic number then a_nx is an algebraic integer, where Template:Mvar satisfies a polynomial p(x) with integer coefficients and where a_nxⁿ is the highest-degree term of p(x). The value y = a_nx is an algebraic integer because it is a root of q(y) = a p(y/a_n), 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 Template:Absx / Template:Abs, where Template:Mvar satisfies a polynomial p(x) with integer coefficients and where a_nxⁿ is the highest-degree term of p(x).
• The only rational algebraic integers are the integers. That is, if Template:Mvar is an algebraic integer and $x\in \mathbb{\mathbb{Q}}$ then $x\in \mathbb{\mathbb{Z}}$. This is a direct result of the rational root theorem for the case of a monic polynomial.

See also

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