In
field theory, a primitive element of a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
is a
generator of the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of the field. In other words, is called a primitive element if it is a
primitive th root of unity in ; this means that each non-zero element of can be written as for some
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
.
If is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, the elements of can be identified with the
integers modulo . In this case, a primitive element is also called a
primitive root modulo .
For example, 2 is a primitive element of the field and , but not of since it generates the cyclic subgroup of order 3; however, 3 is a primitive element of . 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 is , where is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
, which counts the number of elements less than or equal to that are
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
to . This can be proved by using the theorem that the multiplicative group of a finite field is
cyclic of order , and the fact that a finite cyclic group of order contains generators.
See also
*
Simple extension
*
Primitive element theorem
*
Zech's logarithm
References
*
External links
*
Finite fields
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