Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety. The polynomial ring, K[X], in X over a field (or, more generally, a commutative ring) K can be defined in several equivalent ways. One of them is to define K[X] as the set of expressions, called polynomials in X, of the form where p0, p1, ..., pm, the coefficients of p, are elements of K, p_m ≠ 0 if m > 0, and X, X^2, ..., are symbols, which are considered as "powers" of X, and follow the usual rules of exponentiation: X^0 = 1, X^1 = X, and for any nonnegative integers k and l. The symbol X is called an indeterminate or variable. (The term of "variable" comes from the terminology of polynomial functions. However, here, X has not any value (other than itself), and cannot vary, being a constant in the polynomial ring.) Two polynomials are equal when the corresponding coefficients of each X^k are equal. One can think of the ring K[X] as arising from K by adding one new element X that is external to K, commutes with all elements of K, and has no other specific properties.