### 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,

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

2 See also

3 References

4 External links

Properties[edit]

Number of primitive elements[edit]

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[edit]

Simple extension

Primitive root

Zech’s logarithm

References[edit]

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

External links[edit]

Weisstein, Eric W. “Primitive Polynomial”. MathWorld.

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