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. 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. Polynomial rings occur in many parts of mathematics, and the study of their properties was among the main motivations for the development of commutative algebra and ring theory. Polynomial rings and their ideals are fundamental in algebraic geometry. Many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, are generalizations 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, in X over a field K is defined as the set of expressions, called polynomials in X, of the form where p0, p1, ..., pm, the coefficients of p, are elements of K, and X, X2, are symbols, which are considered as 'powers of X', and, by convention, follow the usual rules of exponentiation: X0 = 1, X1 = X, and X k X l = X k + l {displaystyle X^{k},X^{l}=X^{k+l}} for any nonnegative integers k and l. The symbol X is called an indeterminate or variable. Two polynomials are defined to be equal when the corresponding coefficient of each Xk is equal. This terminology is suggested by real or complex polynomial functions. However, in general, X and its powers, Xk, are treated as formal symbols, not as elements of the field K or functions over it. One can think of the ring K as arising from K by adding one new element X that is external to K and requiring that X commute with all elements of K. The polynomial ring in X over K is equipped with an addition, a multiplication and a scalar multiplication that make it a commutative algebra. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if