Primitive Element

Related Concepts

Extension Field; Finite Field; Generator; Group

Definition

A primitive element of a finite field is a generator of the multiplicative group of the field.

Theory

Let α be an element of the finite field ${\text{ F} }_{{q}^{d}}$. If α is a generator of ${\text{ F} }_{{q}^{d}}^{{_\ast}}$, i.e., if the set of elements $\alpha, {\alpha }^{2},{\alpha }^{3},\,\ldots$ traverses all elements in ${\text{ F} }_{{q}^{d}}^{{_\ast}}$, then α is a primitive element.

Let f(x) be the irreducible polynomial of degree d used to construct the extension field $\text{ F }_{{q}^{d}}$ over the subfield F _q , and let α be a root of f(x). If α is a primitive element then f(x) is called a primitive polynomial.

Every finite field ${\text{ F} }_{{q}^{d}}$ has $\phi ({q}^{d} - 1)$ primitive elements and $\phi ({q}^{d} - 1)/d$ primitive polynomials, where ϕ is Euler’s totient function.

In the degenerate case where d = 1, any element α that generates the multiplicative group \({\text{...