Math 2.0

statement theorem

Wedderburn's little theorem

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.

Explanations

L1 human

In an infinite division ring, aba b and bab a can be different. The theorem says that this cannot happen when the ring is finite. So there are no finite analogues of the quaternions.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

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To read this node, read these 7 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 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 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.
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  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

Sources

  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

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