Irreducible Polynomial

Related Concepts

Extension Field; Field; Finite Field

Definition

A polynomial that is not divisible by any smaller polynomials other than trivial ones is an irreducible polynomial.

Theory

Let f(x) be a polynomial

$$f(x) = {f}_{d}{x}^{d} + {f}_{ d-1}{x}^{d-1} + \cdots + {f}_{ 1}x + {f}_{0},$$

where the coefficients ${f}_{0},\ldots, {f}_{d}$ are elements of a field F. If there is another polynomial g(x) over F with degree between 1 and d − 1 such that g(x) divides f(x), then f(x) is reducible. Otherwise, f(x)is irreducible. (Nonzero polynomials of degree 0, i.e., nonzero elements of F, divide every polynomial so they are considered “trivial” factors.)

As an example, the polynomial ${x}^{2} + 1$ over the finite field F ${}_{2}$ is reducible since ${x}^{2} + 1 = {(x + 1)}^{2}$, whereas the polynomial ${x}^{2} + x + 1$ is irreducible.

A representation of the finite field ${\mathbb{F}}_{{q}^{d}}$ can be constructed from a representation of the finite field F ${}_{q}$together...