algebraic integer AlgebraicInteger 2001-10-15 20:16:52 2006-10-04 22:15:01 Definition algebraic number theory \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $K$ be an \PMlinkname{extension}{ExtensionField} of $\mathbb{Q}$ contained in $\mathbb{C}$. A number $\alpha \in K$ is called an \emph{algebraic integer} of $K$ if it is the root of a monic polynomial with coefficients in $\mathbb{Z}$, i.e., an element of $K$ that is integral over $\mathbb{Z}$. Every algebraic integer is an algebraic number (with $K = \mathbb{C}$), but the converse is false.