Math 2.0

Field theory and polynomials Definition

Primitive element of a finite field

Builds on Finite fieldThe multiplicative group of a finite field is cyclic

Definition

Let FF be a finite field of order qq. An element α∈F\alpha \in F is a primitive element of FF if α\alpha generates the multiplicative group F×F^\times. That is, F×={α,α2,…,αq−1=1}F^\times = \{ \alpha, \alpha^2, \dots, \alpha^{q-1} = 1 \}.

Remarks

Primitive elements exist, because F×F^\times is cyclic. There are exactly φ(q−1)\varphi(q - 1) of them, where φ\varphi is Euler's totient function. The element 22 is a primitive element of 𝔽3\mathbb{F}_3 and of 𝔽5\mathbb{F}_5, but not of 𝔽7\mathbb{F}_7. The element 33 is a primitive element of 𝔽7\mathbb{F}_7. A primitive element α\alpha of FF generates FF as a simple extension of its prime subfield. The converse is false in general.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionField, definition, no evidence levelFielddefinitionGroup of units of a ring, definition, no evidence levelGroup of units of a r…definitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionFinite field, definition, no evidence levelFinite fielddefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionMultiplicity of a root, definition, no evidence levelMultiplicity of a rootdefinitionDegree of a product, statement, no evidence levelDegree of a productstatementDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementRemainder theorem, statement, no evidence levelRemainder theoremstatementFactor theorem, statement, no evidence levelFactor theoremstatementA polynomial over a field has at most as many roots as its degree, statement, no evidence levelA polynomial over a f…statementFinite subgroups of the multiplicative group of a field are cyclic, statement, no evidence levelFinite subgroups of t…statementThe multiplicative group of a finite field is cyclic, statement, no evidence levelThe multiplicative gr…statementPrimitive element of a finite field, definition, no evidence levelPrimitive element of …definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Unit of a ring definition
  6. Zero divisor definition
  7. Degree of a polynomial definition
  8. Field definition
  9. Group of units of a ring definition
  10. Polynomial evaluation definition
  11. Integral domain definition
  12. Leading coefficient definition
  13. Finite field definition
  14. Root of a polynomial definition
  15. Monic polynomial definition
  16. Multiplicity of a root definition
  17. Degree of a product statement
  18. Division algorithm for polynomials statement
  19. Remainder theorem statement
  20. Factor theorem statement
  21. A polynomial over a field has at most as many roots as its degree statement
  22. Finite subgroups of the multiplicative group of a field are cyclic statement
  23. The multiplicative group of a finite field is cyclic statement
  24. Primitive element of a finite field definition

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Primitive_element_(finite_field), lead section
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Kinddefinition
WritingW0 stub
MSC12E20
DefinesPrimitive element of a finite field
UsesFinite field, The multiplicative group of a finite field is cyclic

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