Polynomial over a field ( $F i e l d [X]$ )

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,

G P (3)

and with elements

{0, 1, 2}

, we can denote polynomials over that ring as

G P (3) [x]

where

x

is the variable name.

For example, one such polynomial could be:

P (x) = 2 x^{4} + x^{2} + 2

and another one:

Q (X) = x^{3} + 2 x^{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

and so on.

We can also add polynomials as usual over the field:

P (x) + Q (x) = 2 x^{4} + x^{3} + (1 + 2) x^{2} + (2 + 2) = 2 x^{4} + x^{3} + (0) x^{2} + 1 = 2 x^{4} + x^{3} + 1

and multiplication works analogously.

 Ancestors