Polynomial ring
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, can be defined in several equivalent ways. One of them is to define K as the set of expressions, called polynomials in X, of the form

where p, p, …, p, the coefficients of p, are elements of K, p ≠ 0 if m > 0, and X, X, …, are symbols, which are considered as "powers" of X, and follow the usual rules of exponentiation: X = 1, X = 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. (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 X are equal.

One can think of the ring K 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

and

then

and

where k = max(m, n), l = m + n,

and

In these formulas, the polynomials p and q are extended by adding "dummy terms" with zero coefficients, so that all p and q that appear in the formulas are defined. Specifically, if m < n, then p = 0 for m < i ≤ n.

The scalar multiplication is the special case of the multiplication where p = p is reduced to its constant term (the term that is independent of X); that is

It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over K. Therefore, polynomial rings are also called polynomial algebras.

Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite sequence (p, p, p, …) of elements of K, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some m so that p = 0 for n > m. In this case, p and X are considered as alternate notations for the sequences (p, 0, 0, …) and (0, 1, 0, 0, …), respectively. A straightforward use of the operation rules shows that the expression

is then an alternate notation for the sequence
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