Integral element

Summary

An element $b$ of a commutative ring $B$ is integral over a subring $A$ if it is a root of a monic polynomial over $A$. It extends to rings the notion of an algebraic element of a field extension, with the extra demand that the leading coefficient is $1$. Over $\Z$, the integral elements of $\C$ are the algebraic integers, the basic objects of algebraic number theory. Integral elements are the starting point of the theory of integral extensions, integral closure and normal rings.