{title2=$Field[X]$}
{wiki}

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 (algebra)> 9, $GP(3)$ and with elements $\{0, 1, 2\}$, we can denote polynomials over that ring as
$$
GP(3)[x]
$$
where $x$ is the variable name.

For example, one such polynomial could be:
$$
P(x) = 2x^4 + x^2 + 2
$$
and another one:
$$
Q(X) = x^3 + 2x^2 + 2
$$
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.:
$$
P(0) = 2 (0 \times 0 \times 0 \times 0) + (0 \times 0) + 2 = 2
P(1) = 2 (1 \times 1 \times 1 \times 1) + (1 \times 1) + 2 = 2 (1) + 1 + 2 = 2
P(2) = 2 (2 \times 2 \times 2 \times 2) + (2 \times 2) + 2 = 2 (16 % 3) + (4 % 3) + 2 = 2 + 1 + 2 = 2
$$
and so on.

We can also add polynomials as usual over the field:
$$
P(x) + Q(x) = 2x^4 + x^3 + (1+2)x^2 + (2 + 2) = 2x^4 + x^3 + (0)x^2 + 1 = 2x^4 + x^3 + 1
$$
and multiplication works analogously.


Polynomial over a field