Polynomial

algebra

Definition

Polynomial

Let $F$ be a field. A polynomial in one indeterminate $x$ over $F$ is a formal expression

$$p(x)=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n,$$

where $a_i\in F$ and only finitely many $a_i$ are non-zero.

Equivalently, $p$ is a sequence $(a_0,a_1,a_2,\dots )$ in $F$ with finite support. The set of all such polynomials is denoted $F[x]$ and forms a polynomial ring under pointwise addition and Cauchy-product multiplication.