Quantum no-deleting theorem: Difference between revisions

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In [[field theory (mathematics)|field theory]], a '''primitive element''' of a [[finite field]] ''GF''(''q'') is a [[generating set of a group|generator]] of the [[group of units|multiplicative group]] of the field. In other words, <math>\alpha \in \mathrm{GF}(q)</math> is called a primitive element if it is a [[primitive root of unity|primitive (''q''-1) root of unity]] in ''GF''(''q''); this means that all the non-zero elements of <math>\mathrm{GF}(q)</math> can be written as <math>\alpha^i</math> for some (positive) integer <math>i</math>.


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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 (field theory)|minimal polynomial]] of a primitive element is a [[primitive polynomial (field theory)|primitive polynomial]].
 
==Properties==
 
===Number of Primitive Elements===
The number of primitive elements in a finite field ''GF''(''n'') is ''φ''(''n'' - 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''(''n'') is [[Field_(mathematics)#Some_first_theorems|cyclic]] of order ''n'' - 1, and the fact that a finite cyclic group of order ''m'' contains ''φ''(''m'') generators.
 
==See also==
* [[Primitive element (field theory)]]
* [[Primitive root]]
 
==References==
* {{cite book | last=Lidl | first=Rudolf | coauthors=Harald Niederreiter | title=Finite Fields | edition=2nd | year=1997 | publisher=[[Cambridge University Press]] | isbn=0-521-39231-4 }}
 
==External links==
*{{MathWorld | title=Primitive Polynomial | urlname=PrimitivePolynomial }}
 
[[Category:Field theory]]
 
 
{{Abstract-algebra-stub}}

Revision as of 04:38, 4 February 2014

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∈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 GF(q) can be written as αi for some (positive) integer 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.

Properties

Number of Primitive Elements

The number of primitive elements in a finite field GF(n) is φ(n - 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(n) is cyclic of order n - 1, and the fact that a finite cyclic group of order m contains φ(m) generators.

See also

References

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