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 all 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.

See also

• Primitive element (field theory)
• Primitive root

References

• Lidl, Rudolf (1997). Finite Fields (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

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