Polynomial Function on a Field

Let $K$ be a field.

A map $p:K\to K$ is a polynomial function on $K$ if there are a natural number $N$, an element $c_{0}\in K$, and a map $c:[N]\to K$ on the initial segment determined by $N$, with values written $c_{k}$, such that

$$p(x)=c_{0}+\sum_{k=1}^{N}c_{k}\,x^{k}\qquad\text{for every }x\in K,$$

where the sum is the finite sum of $K$ and $x^{k}$ is the $k$th power of $x$.

Such a triple $(N,c_{0},c)$ is a system of coefficients for $p$.