Math 2.0

Commutative algebra Theorem

A finite integral domain is a field

Builds on Integral domainField

Statement

Let RR be an integral domain that has a finite number of elements. Then RR is a field.

Remarks

The proof: for a≠0a \neq 0, the map x↦axx \mapsto a x from RR to RR is injective by cancellation. A finite set has no injective map to itself that is not surjective, so ax=1a x = 1 for some xx. A consequence is that ℤ/pℤ\mathbb{Z}/p\mathbb{Z} is a field for every prime pp. The hypothesis of finiteness is necessary: ℤ\mathbb{Z} is an infinite integral domain that is not a field.

Reading path

To understand this page from the ground up, read these 8 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 ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionA finite integral domain is a field, statement, no evidence levelA finite integral dom…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. Zero divisor definition
  6. Field definition
  7. Integral domain definition
  8. A finite integral domain is a field statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral_domain, Properties
About this page
Tag0EBE031B8E
URIm2://0EBE031B8E
Kindstatement, theorem
WritingW0 stub
MSC13G05
StatusThis copy of the site has no build data. Run m2 status 0EBE031B8E.
UsesIntegral domain, Field

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by