Math 2.0

statement corollary

The multiplicative group of a finite field is cyclic

Statement

Let FF be a finite field of order qq. Then the multiplicative group F×F^\times is cyclic of order q−1q - 1.

Remarks

This is the case G=F×G = F^\times of the theorem on finite subgroups. A generator of F×F^\times is a primitive element of FF. For F=Z/pZF = \mathbb{Z}/p\mathbb{Z}, the statement says that there is a primitive root modulo pp. For example, 33 generates (Z/7Z)×(\mathbb{Z}/7\mathbb{Z})^\times, with powers 3,2,6,4,5,13, 2, 6, 4, 5, 1.

Explanations

L1 human

All the nonzero elements of a finite field are powers of one element. So multiplication in a finite field is like addition of exponents modulo q−1q - 1.

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What this needs

To read this node, read these 23 nodes in this order. The foundations come first, and the list ends with this node.

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…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

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Sources

  1. Web https://en.wikipedia.org/wiki/Finite_field, Multiplicative structure

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