Primitive element (finite field)#Number of primitive elements
{{short description|Generator of the multiplicative group of a finite field}}
{{distinguish|text = primitive element of a simple extension}}
In field theory, a primitive element of a finite field {{math|GF(q)}} is a generator of the multiplicative group of the field. In other words, {{math|α ∈ GF(q)}} is called a primitive element if it is a primitive root of unity in {{math|GF(q)}}; this means that each non-zero element of {{math|GF(q)}} can be written as {{math|α{{i sup|i}}}} for some natural number {{math|i}}.
If {{mvar|q}} is a prime number, the elements of {{math|GF(q)}} can be identified with the integers modulo n. In this case, a primitive element is also called a Primitive root modulo n.
For example, 2 is a primitive element of the field {{math|GF(3)}} and {{math|GF(5)}}, but not of {{math|GF(7)}} since it generates the cyclic subgroup {{math|1={2, 4, 1} }} of order 3; however, 3 is a primitive element of {{math|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 {{math|GF(q)}} is {{math|φ(q − 1)}}, where {{math|φ}} is Euler's totient function, which counts the number of elements less than or equal to {{math|m}} that are coprime to {{math|m}}. This can be proved by using the theorem that the multiplicative group of a finite field {{math|GF(q)}} is cyclic of order {{math|q − 1}}, and the fact that a finite cyclic group of order {{math|m}} contains {{math|φ(m)}} generators.
See also
References
- {{cite book | last=Lidl | first=Rudolf |author2=Harald Niederreiter |author2-link= Harald Niederreiter | title=Finite Fields | url=https://archive.org/details/finitefields0000lidl_a8r3 | url-access=registration | edition=2nd | year=1997 | publisher=Cambridge University Press | isbn=0-521-39231-4 }}
External links
- {{MathWorld | title=Primitive Polynomial | urlname=PrimitivePolynomial }}
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