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 (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers $A$ is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

This article is about the ring of complex numbers integral over

\mathbb {Z}

. For the general notion of algebraic integer, see Integrality.

The ring of integers of a number field $K$ , denoted by $O K$ , 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 $α$ is an algebraic integer if and only if the ring $\mathbb {Z} [\alpha ]$ is finitely generated as an abelian group, which is to say, as a $\mathbb {Z}$ -module.

$α \in K$ is an algebraic integer if there exists a monic polynomial $f(x)\in \mathbb {Z} [x]$ such that $f (α) = 0$ .

$α \in K$ is an algebraic integer if the monic polynomial of $α$ over $\mathbb {Q}$ is in $\mathbb {Z} [x]$ .

minimal

$α \in K$ is an algebraic integer if $\mathbb {Z} [\alpha ]$ is a finitely generated $\mathbb {Z}$ -module.

$α \in K$ is an algebraic integer if there exists a non-zero finitely generated $\mathbb {Z}$ - $M\subset \mathbb {C}$ such that $αM \subseteq M$ .

submodule

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

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 $K/\mathbb {Q}$ .

The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of $\mathbb {Q}$ and $A$ is exactly $\mathbb {Z}$ . The rational number $a / b$ is not an algebraic integer unless $b$ $a$ . The leading coefficient of the polynomial $bx - a$ is the integer $b$ .

divides

The ${\sqrt {n}}$ of a nonnegative integer $n$ is an algebraic integer, but is irrational unless $n$ is a perfect square.

square root

If $d$ is a then the extension $K=\mathbb {Q} ({\sqrt {d}}\,)$ is a quadratic field of rational numbers. The ring of algebraic integers $O K$ contains ${\sqrt {d}}$ since this is a root of the monic polynomial $x 2 - d$ . Moreover, if $d \equiv 1 mod 4$ , then the element ${\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}$ is also an algebraic integer. It satisfies the polynomial $x 2 - x + 1 / 4 (1 - d)$ where the constant term $1 / 4 (1 - d)$ is an integer. The full ring of integers is generated by ${\sqrt {d}}$ or ${\textstyle {\frac {1}{2}}(1+{\sqrt {d}}\,)}$ respectively. See Quadratic integer for more.

square-free integer

The ring of integers of the field $F=\mathbb {Q} [\alpha ]$ , $α = 3 \sqrt m$ , has the following , writing $m = hk 2$ for two square-free coprime integers $h$ and $k$ :^[1]
${\begin{cases}1,\alpha ,{\dfrac {\alpha ^{2}\pm k^{2}\alpha +k^{2}}{3k}}&m\equiv \pm 1{\bmod {9}}\\1,\alpha ,{\dfrac {\alpha ^{2}}{k}}&{\text{otherwise}}\end{cases}}$

integral basis

If $ζ n$ is a $n$ th root of unity, then the ring of integers of the cyclotomic field $\mathbb {Q} (\zeta _{n})$ is precisely $\mathbb {Z} [\zeta _{n}]$ .

primitive

If $α$ is an algebraic integer then $β = n \sqrt α$ is another algebraic integer. A polynomial for $β$ is obtained by substituting $x n$ in the polynomial for $α$ .

If $P (x)$ is a that has integer coefficients but is not monic, and $P$ is irreducible over $\mathbb {Q}$ , then none of the roots of $P$ are algebraic integers (but are algebraic numbers). Here primitive is used in the sense that the highest common factor of the coefficients of $P$ is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.

primitive polynomial

Finite generation of ring extension[edit]

For any $α$ , the ring extension (in the sense that is equivalent to field extension) of the integers by $α$ , denoted by $\mathbb {Z} (\alpha )\equiv \{\sum _{i=0}^{n}\alpha ^{i}z_{i}|z_{i}\in \mathbb {Z} ,n\in \mathbb {Z} \}$ , 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 $\mathbb {Q}$ there replaced by $\mathbb {Z}$ here, and the notion of field extension degree replaced by finite generation (using the fact that $\mathbb {Z}$ 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 $\mathbb {Z}$ or $\mathbb {Q}$ , respectively.

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 are not. This is the Abel–Ruffini theorem.

quintics

The ring of algebraic integers is a , as a consequence of the principal ideal theorem.

Bézout domain

If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the 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.

reciprocal

If $x$ is an algebraic number then $a n x$ is an algebraic integer, where $x$ satisfies a polynomial $p (x)$ with integer coefficients and where $a n x n$ is the highest-degree term of $p (x)$ . The value $y = a n x$ is an algebraic integer because it is a root of $q (y) = a n - 1 n p (y / a n)$ , 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 $| a n | x / | a n |$ , where $x$ satisfies a polynomial $p (x)$ with integer coefficients and where $a n x n$ is the highest-degree term of $p (x)$ .

The only rational algebraic integers are the integers. Thus, if $α$ is an algebraic integers and $\alpha \in \mathbb {Q}$ , then $\alpha \in \mathbb {Z}$ . This is a direct result of the for the case of a monic polynomial.

Algebraic integer

minimal

submodule

divides

square root

square-free integer

integral basis

primitive

primitive polynomial

Finite generation of ring extension[edit]

quintics

Bézout domain

reciprocal

rational root theorem

Integral element

Gaussian integer

Eisenstein integer

Root of unity

Dirichlet's unit theorem

Fundamental units