Vector Space of Polynomials

Definition

Suppose $\F $ is a field. A function $p: \F \to \F $ is called a polynomial with coefficients in $\F $ if there exist $a_0, a_1, \dots , a_m \in \F $ (for some $m \in \N $) such that

\[ p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_mz^m \quad \text{for all } z \in \F \]

The set of all polynomials with coefficients in $\F $ is a subspace of the vector space of all functions $\F ^\F $.

Notation

The set of all polynomials in $\F $ is sometimes denoted by $\mathbfsf{P} (\F )$ or $\mathcal{P} (\F )$.