Polynomial over a field ($Field[X]$)

By default, we think of polynomials over the real numbers or complex numbers.

However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.

For example, given the finite field of order 9, $GP(3)$ and with elements $\{0,1,2\}$, we can denote polynomials over that ring aswhere $x$ is the variable name.

For example, one such polynomial could be:and another one:Note how all the coefficients are members of the finite field we chose.

Given this, we could evaluate the polynomial for any element of the field, e.g.:and so on.

We can also add polynomials as usual over the field:and multiplication works analogously.