#math/linear-algebra A function $p:\mathbf{F}\to \mathbf{F}$ is called a `polynomial` over $\mathbf{F}$ if there exists $a_{0}, \dots, a_{n}\in \mathbf{F}$ such that $ p(z) = a_{0}+a_{1}z+\dots+a_{n}z^n $ for all $Z\in \mathbf{F}$. We define $\mathcal{P}(\mathbf{F})$ as the set of all polynomials with coefficients in $\mathbf{F}$. For a particular polynomial, the degree is defined as the largest $i$ where $a_{i}\neq{}0$. $\mathcal{P}_{m}(\mathbf{F})=$ span$(1, z, \dots, z^m)$ where $z^k$ is the function $f_{k}(x)=x^k$ for all $x \in \mathbf{F}$. Then, $\mathcal{P}_{m}(\mathbf{F})$ is the set of all polynomials, $p$, where $\text{deg }p\leq{}m$. > [!Edge case] > The $0_{v}$ polynomial has degree $-\infty$.