Primitive element (finite field)

In field theory, a primitive element of a finite field $GF(q)$ is a generator of the multiplicative group of the field. In other words, $α \in GF(q)$ is called a primitive element if it is a primitive $(q - 1)$ th root of unity in $GF(q)$ ; this means that each non-zero element of $GF(q)$ can be written as $α i$ for some natural number $i$ .

If $q$ is a prime number, the elements of $GF(q)$ can be identified with the integers modulo $q$ . In this case, a primitive element is also called a primitive root modulo $q$ .

For example, 2 is a primitive element of the field $GF(3)$ and $GF(5)$ , but not of $GF(7)$ since it generates the cyclic subgroup ${2, 4, 1}$ of order 3; however, 3 is a primitive element of $GF(7)$ . The minimal polynomial of a primitive element is a primitive polynomial.

Properties

Number of primitive elements

The number of primitive elements in a finite field $GF(q)$ is $φ (q - 1)$ , where $φ$ is Euler's totient function, which counts the number of elements less than or equal to $m$ that are coprime to $m$ . This can be proved by using the theorem that the multiplicative group of a finite field $GF(q)$ is cyclic of order $q - 1$ , and the fact that a finite cyclic group of order $m$ contains $φ (m)$ generators.