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 \(\mathbf {A} \cap \mathbf {Q} = \mathbf {Z}\).