### Primitive element (finite field)

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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

G
F

(
q
)

{\displaystyle \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

G
F

(
q
)

{\displaystyle \mathrm {GF} (q)}

can be written as

α

i

{\displaystyle \alpha ^{i}}

for some (positive) integer

i

{\displaystyle 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.

Contents

1 Properties

1.1 Number of primitive elements

3 References

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.

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.