# 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* ∈ 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 integer *i*.

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[edit]

### Number of primitive elements[edit]

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* which are relatively prime 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.

## See also[edit]

## References[edit]

- Lidl, Rudolf; Harald Niederreiter (1997).
*Finite Fields*(2nd ed.). Cambridge University Press. ISBN 0-521-39231-4.

## External links[edit]

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