Polynomring

Polynomring, or the polynomial ring, in one indeterminate over a ring R is the set of polynomials with coefficients in R:

R[X] = {a0 + a1 X + a2 X^2 + ... + an X^n | n ≥ 0, ai ∈ R}.

Here X is an indeterminate, and addition and multiplication are the usual operations obtained by treating X^i as formal symbols and using the distributive law.

If R is commutative with 1, then R[X] is a commutative ring with unity. Its structural properties reflect those of R: for example, if R is a field, R[X] is a principal ideal domain and a unique factorization domain; if R is a general commutative ring, R[X] is still a commutative ring, and details like whether it is a domain depend on R.

Key notions include the degree of a nonzero polynomial f = a0 + a1 X + ... + an X^n, which is n, and the leading coefficient, lc(f) = an. The units of R[X] are precisely the constant polynomials whose coefficient is a unit of R. The product rule deg(fg) = deg f + deg g holds for nonzero polynomials.

Ideals in R[X] are often principal, generated by a polynomial f. When R is a field, irreducible polynomials in R[X] play a central role in factorization and in constructing field extensions.

Extensions and generalizations include multivariate polynomial rings R[x1, ..., xn], where polynomials use several indeterminates. Substitution homomorphisms, evaluation at a ∈ R or in an R-algebra, yield natural ring homomorphisms R[X] → S.

Examples include the rings Z[X] and F[X] for a field F. Polynomial rings form a fundamental tool in algebra, number theory, algebraic geometry, and computer algebra.