Math 2.0

Field theory and polynomials Finite fields Theorem

The order of a finite field is a prime power

Builds on Finite fieldThe characteristic of an integral domain is zero or primePrime subfieldDegree of a field extension

Statement

Let FF be a finite field. Then the characteristic of FF is a prime number pp, and ∣F∣=pn|F| = p^n with n=[F:𝔽p]n = [F : \mathbb{F}_p].

Remarks

The proof: a finite field has nonzero characteristic, and the characteristic of a domain is 00 or prime. The prime subfield is 𝔽p\mathbb{F}_p, and FF is a vector space of dimension nn over it, so it has pnp^n elements. So there is no field with 66 or 1010 elements. The converse, that a field of order pnp^n exists for each n≥1n \ge 1, is the existence theorem.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCharacteristic of a ring, definition, no evidence levelCharacteristic of a r…definitionCommutative ring, definition, no evidence levelCommutative ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit 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 domaindefinitionFinite field, definition, no evidence levelFinite fielddefinitionThe characteristic of an integral domain is zero or prime, statement, no evidence levelThe characteristic of…statementSubfield, definition, no evidence levelSubfielddefinitionPrime subfield, definition, no evidence levelPrime subfielddefinitionField extension, definition, no evidence levelField extensiondefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionThe order of a finite field is a prime power, statement, no evidence levelThe order of a finite…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.
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  1. Group concept
  2. Ring definition
  3. Characteristic of a ring definition
  4. Commutative ring definition
  5. Subring definition
  6. Unit of a ring definition
  7. Zero divisor definition
  8. Field definition
  9. Integral domain definition
  10. Finite field definition
  11. The characteristic of an integral domain is zero or prime statement
  12. Subfield definition
  13. Prime subfield definition
  14. Field extension definition
  15. Degree of a field extension definition
  16. The order of a finite field is a prime power statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_field, Properties
About this page
Tag0H5WWC8RZX
URIm2://0H5WWC8RZX
Kindstatement, theorem
WritingW0 stub
MSC12E20
StatusThis copy of the site has no build data. Run m2 status 0H5WWC8RZX.
UsesFinite field, The characteristic of an integral domain is zero or prime, Prime subfield, Degree of a field extension

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