For questions regarding algebraic integers, which is a complex number which is integral over the integers.
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.
The following are equivalent definitions of an algebraic integer. Let K be a number field (i.e., a finite extension of $\displaystyle \mathbb {Q} $, the field of rational numbers), in other words, $\displaystyle K=\mathbb {Q} (\theta )$ for some algebraic number $\displaystyle \theta \in \mathbb {C}$ by the primitive element theorem.
- $α ∈ K$ is an algebraic integer if there exists a monic polynomial $\displaystyle f(x)\in \mathbb {Z} [x]$ such that $f(α) = 0$.
- $α ∈ K$ is an algebraic integer if the minimal monic polynomial of α over $\displaystyle \mathbb {Q} $ is in $\displaystyle \mathbb {Z} [x]$.
- $α ∈ K$ is an algebraic integer if $\displaystyle \mathbb {Z} [\alpha ]$ is a finitely generated $\displaystyle \mathbb {Z} $-module.
- $α ∈ K$ is an algebraic integer if there exists a non-zero finitely generated $\displaystyle \mathbb {Z} $-submodule $\displaystyle 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 $\displaystyle K/\mathbb {Q}$.
