Polynomial ring

Let $R$ be a commutative ring with $1$. The polynomial ring $R[x]$ consists of finite sums $\sum_{i=0}^n a_i x^i$ with coefficients $a_i\in R$, with addition termwise and multiplication determined by distributivity and $x^i x^j=x^{i+j}$.

Polynomial rings provide the basic algebraic enlargement of a ring by adjoining an indeterminate, and they interact strongly with divisibility and factorization (e.g. via irreducible polynomials ). The coefficient data of a polynomial is often packaged by its content .

Examples:

• $\mathbb{Z}[x]$ is the ring of integer-coefficient polynomials.
• If $k$ is a field, then $k[x,y]$ can be viewed as $(k[x])[y]$, polynomials in $y$ with coefficients in $k[x]$.
• The polynomial $f(x)=2x^2-3x+1\in \mathbb{Z}[x]$ has degree $2$ and leading coefficient $2$.