Math 2.0

Field theory and polynomials Finite fields Theorem

Wedderburn's little theorem

Builds on Division ringField

Statement

Let DD be a division ring with a finite number of elements. Then the multiplication of DD is commutative, so DD is a field.

Remarks

The hypothesis of finiteness is necessary: the quaternions are an infinite division ring that is not commutative. A consequence is that every finite domain, commutative or not, is a field.

History

Joseph Wedderburn gave the first proof in 1905. Leonard Eugene Dickson gave another proof soon after. The first proof of Wedderburn had a gap, and for this reason some historians credit Dickson with the first correct proof.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionField, definition, no evidence levelFielddefinitionWedderburn's little theorem, statement, no evidence levelWedderburn's little 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. Unit of a ring definition
  5. Division ring definition
  6. Field definition
  7. Wedderburn's little theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem, History
About this page
Tag0RRNVX1VKA
URIm2://0RRNVX1VKA
Kindstatement, theorem
WritingW1 stated
MSC12E15
StatusThis copy of the site has no build data. Run m2 status 0RRNVX1VKA.
UsesDivision ring, Field
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  • explanation L2 (0 of 1)

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