IDA: Interactive Document on Algebra

Let R be a ring and let X be an indeterminate. By R[X] we denote the set of all polynomials in X with coefficients in R, compare Section 3.1.

Let

a = a₀ + a₁X + ··· + a_nXⁿ and b = b₀ + b₁X + ··· + b_mX^m

be two elements of R[X]. By adding, if necessary, some terms 0X^k we may assume n = m.

The sum of these polynomials is

a + b = (a₀ + b₀) + (a₁ + b₁)X + ··· + (a_n + b_n)Xⁿ.

The product of these polynomials is

a * b = c₀ + c₁X + ··· + c_{n + m}X^{n+ m},

where c_k = a₀b_k + a₁b_{k - 1} + ··· + a_kb₀.

The symbol * is often omitted.

Theorem

The sum and product of polynomials define a ring structure on the set R[X] of all polynomials in X with coefficients in R. The zero element is the zero polynomial 0; the unit element is the polynomial 1.

The ring R[X] is called the polynomial ring over R in the indeterminate X. The ring R is called the coefficient ring of R[X].