Algebraic integer - Knowlpedia

An algebraic number $\alpha\in\overline{\mathbb Q}$ is an algebraic integer if it is integral over $\mathbb Z$ : there are $n\ge1$ and $a_0,\ldots,a_{n-1}\in\mathbb Z$ with

\alpha^n+a_{n-1}\alpha^{n-1}+\cdots+a_0=0.

The leading coefficient must be one.

Examples

Both $i$ and $\sqrt2$ are algebraic integers, using $T^2+1$ and $T^2-2$ . The rational number $1/2$ is algebraic but not integral: clearing a monic equation of degree $n$ would make an odd integer equal to an even integer.

Rational special case

More generally, if $a/b$ is in lowest terms with $b>0$ , a monic integral equation forces $b\mid a^n$ , hence $b=1$ . Thus the rational algebraic integers are exactly $\mathbb Z$ .

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §2, “Integral elements,” especially Proposition 2.11.