leading coefficient; so, if the polynomial is monic, then and the number is an integer. Conversely, an integer is a root of the monic polynomial

It can be proved that, if two elements of a field are integral over a subring of , then the sum and the product of these elements are also integral over . It follows that the elements of that are integral over form a ring, called the integral closure of in . An integral domain that equals its integral closure in its field of fractions