In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring 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.
Definition (univariate case) Let K be a field or (more generally) a commutative ring. The polynomial ring in X over K, which is denoted K[X], 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
p = p 0 + p 1 X + p 2 X 2 + ⋯ + p m − 1 X m − 1 + p m X m , {\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}
where m is a nonnegative integer, the coefficients p0, p1, ..., pm of p are elements of K, and X, X2, …, are symbols called "powers" of X that follow the usual rules of exponents: X0 = 1, X1 = X, and X k ⋅ X l = X k + l {\displaystyle X^{k}\cdot X^{l}=X^{k+l}} 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 no value (other than itself), and cannot vary, being a constant in the polynomial ring.) Two polynomials are equal when the corresponding coefficients of each Xk 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. This can be used for an equivalent definition of polynomial rings. 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
p = p 0 + p 1 X + p 2 X 2 + ⋯ + p m X m , {\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m},}
and
q = q 0 + q 1 X + q 2 X 2 + ⋯ + q n X n , {\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}
then
p + q = r 0 + r 1 X + r 2 X 2 + ⋯ + r k X k , {\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}
and
p q = s 0 + s 1 X + s 2 X 2 + ⋯ + s l X l , {\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}
where k = max(m, n), l = m + n,
r i = p i + q i {\displaystyle r_{i}=p_{i}+q_{i}}
and
… excerpt ends here. Continue reading the full article.
