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, $\alpha \in \mathrm {GF} (q)$ 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 $\mathrm {GF} (q)$ can be written as $\alpha ^{i}$ for some (positive) 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 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(q) is φ(q - 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(q) is cyclic of order q - 1, and the fact that a finite cyclic group of order m contains φ(m) generators.

See also

• Simple extension
• Primitive root
• Zech's logarithm

References

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

External links

• Weisstein, Eric W. "Primitive Polynomial". MathWorld.