nLab
polynomial

Given a commutative ring $R$, the set of polynomials in one variable with coefficients in $R$ consists of all formal expressions

$$a_n{z}^n+\cdots +a_{1}z+a_0$$a_n z^n + \cdots + a_1 z + a_0

where $n$ is an arbitrary natural number and $a_0,\dots ,a_n\in R$, modulo the equivalence relation generated by equations of the form

$$0z^{n+1}+a_n{z}^n+\cdots +a_{1}z+a_0=a_n{z}^n+\cdots +a_{1}z+a_0$$0 z^{n+1} + a_n z^n + \cdots + a_1 z + a_0 = a_n z^n + \cdots + a_1 z + a_0

(so that we ignore coefficients of zero).

These polynomials also form a commutative ring, denoted $R[z]$, which is in fact a commutative algebra over $R$; the addition and multiplication in this ring are defined in the obvious way. The ring $R[z]$ may also be described more abstractly as the free $R$-algebra on one generator. Similarly, the set of polynomials in any give set of variables with coefficients in $R$ is the free commutative $R$-algebra on that set of generators; see symmetric power and symmetric algebra.

The field of fractions of $R[z]$ is the field $R(z)$ of rational functions.