Algebraic Number Fields

Abstract

A complex number $\xi$ is called an algebraic integer if $\mathbf {Z}[ \xi ]$ is a finitely generated $\mathbf {Z}$-module; this condition is equivalent to the fact that $f( \xi )=0$ for some polynomial $f(X)=X^m+a_1X^{m-1}+ \cdots +a_m$, $a_i \in \mathbf {Z}$. Let $\mathbf {A}$ be the set of all algebraic integers. Then $\mathbf {A}$ is a subring of $\mathbf {C}$, and $\mathbf {A} \cap \mathbf {Q} = \mathbf {Z}$.