Math 2.0

Field theory and polynomials Galois theory Theorem

Finite subgroups of the multiplicative group of a field are cyclic

Builds on Group of units of a ringA polynomial over a field has at most as many roots as its degree

Statement

Let KK be a field and let GG be a finite subgroup of the multiplicative group K×K^\times. Then GG is cyclic.

Remarks

The proof: for each dd, the elements g∈Gg \in G with gd=1g^d = 1 are roots of xd−1x^d - 1. So there are at most dd of them, by the bound on the number of roots, and a finite group with this property is cyclic. For K=ℂK = \mathbb{C}, the finite subgroups of ℂ×\mathbb{C}^\times are the groups of the nn-th roots of unity. The statement fails for division rings: the quaternion group {±1,±i,±j,±k}\{\pm 1, \pm i, \pm j, \pm k\} is a finite subgroup of the nonzero quaternions that is not cyclic.

Reading path

To understand this page from the ground up, read these 20 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 polynomialdefinitionGroup 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 coefficientdefinitionRoot 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…statement
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. Group of units of a ring definition
  9. Polynomial evaluation definition
  10. Integral domain definition
  11. Leading coefficient definition
  12. Root of a polynomial definition
  13. Monic polynomial definition
  14. Multiplicity of a root definition
  15. Degree of a product statement
  16. Division algorithm for polynomials statement
  17. Remainder theorem statement
  18. Factor theorem statement
  19. A polynomial over a field has at most as many roots as its degree statement
  20. Finite subgroups of the multiplicative group of a field are cyclic statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_field, Multiplicative structure
About this page
Tag0CKGBEJX6B
URIm2://0CKGBEJX6B
Kindstatement, theorem
WritingW0 stub
MSC12E20
StatusThis copy of the site has no build data. Run m2 status 0CKGBEJX6B.
UsesGroup of units of a ring, A polynomial over a field has at most as many roots as its degree

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