nLab
polynomial

Definition

Let $R$ be a commutative ring.

Example

The set of polynomials in one variable with coefficients in $R$ is the set $R[\mathbb{N}]$ of all formal linear combinations on elements $n\in \mathbb{N}$, thought of as powers $x^n$ of the variable $x$

$$a_n{z}^n+\cdots +a_{1}z+a_0,$$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).

This set is equipped with the structure of a ring itself, in fact a commutative algebra over $R$, denoted $R[z]$ and called the polynomial ring or ring of polynomials given by the unique bilinear map

$$R[z]\cdot R[z]\to R[z]$$R[z] \cdot R[z] \to R[z]

which on monomials is given by

$$z^k\cdot z^l=z^{k+l}.$$z^k \cdot z^l = z^{k+l} \,.

Properties

Proposition

The polynomial ring $R[z]$ is the free $R$-algebra on one generator (the variable $z$).

Proof

By the definition of free objects one needs to check that ring homomorphisms

$$f:R[z]\to K$$f : R[z] \to K

to another ring K are in natural bijection with functions of sets

$$\overline{f}:*\to K$$\bar f : * \to K

from the singleton to the set underlying $K$. Take $\overline{f}≔f(z)$. Using $R$-linearity, this is directly seen to yield the desired bijection.

Remark

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.

Remark

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

Related concepts

• power series

• monic polynomial

• tensor algebra

• affine line