A monic polynomial is a single-variable polynomial where the leading coefficient, the coefficient of the term with the highest degree, is equal to one. This property simplifies many algebraic operations and is particularly useful in polynomial division and factorization problems.
An integrally closed domain is a commutative ring in which every element that is a root of a monic polynomial with coefficients from the ring is already an element of the ring itself. This property ensures that the ring is as 'complete' as possible with respect to the integral closure, meaning it contains all elements that should belong to it based on polynomial equations with coefficients in the ring.
The integral closure of a ring R in an extension ring S is the set of elements in S that are roots of monic polynomials with coefficients in R. It provides a way to extend the notion of 'closeness' or 'completeness' of a ring, similar to how real numbers complete the rational numbers.