Definition:Polynomial over Ring as Sequence

Definition
Let $R$ be a commutative ring with unity.

A polynomial over $R$ can be defined as a sequence in $R$ whose domain is $\N$ and which has finite support.

That is, it is an element of the ring of sequences of finite support.

Because a field is a commutative ring with unity, this applies in particular to a field.

Also see

 * Definition:Polynomial (Abstract Algebra)