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 that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
is a
generator
Generator may refer to:
* Signal generator, electronic devices that generate repeating or non-repeating electronic signals
* Electric generator, a device that converts mechanical energy to electrical energy.
* Generator (circuit theory), an eleme ...
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 referred to ...
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 integer .
If is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
, 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 which are relatively prime to . This can be proved by using the theorem that the multiplicative group of a finite field is
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
of order , and the fact that a finite cyclic group of order contains generators.
See also
*
Simple extension In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified.
The primitive element theorem provides a characterization ...
*
Primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the exten ...
*
Zech's logarithm
Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha.
Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl G. J. Jacobi who used ...
References
*
External links
*
Finite fields
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