# 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, is called a primitive element if it is a primitive (*q*-1) root of unity in *GF*(*q*); this means that all the non-zero elements of can be written as for some (positive) integer .

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 of order 3 {2,4,1}; 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*(*n*) is *φ*(*n* - 1), where *φ*(*m*) 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*(*n*) is cyclic of order *n* - 1, and the fact that a finite cyclic group of order *m* contains *φ*(*m*) generators.

## See also

## References

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