Primitive element (finite field)
In
primitive (q − 1)th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as αi for some natural number
i.
If q is a
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
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.
See also
References
- Lidl, Rudolf; ISBN 0-521-39231-4.
External links
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