Math 2.0

Field theory and polynomials Theorem

Subfields of a finite field

Builds on Finite fieldTower lawEvery element of a finite field of order q satisfies x^q = xSubfield

Statement

Let pp be a prime number and let n≥1n \ge 1. For each divisor mm of nn, the field 𝔽pn\mathbb{F}_{p^n} has exactly one subfield of order pmp^m, namely {a:apm=a}\{ a : a^{p^m} = a \}. It has no other subfields.

Remarks

If KK is a subfield of 𝔽pn\mathbb{F}_{p^n} of order pmp^m, then n=m⋅[𝔽pn:K]n = m \cdot [\mathbb{F}_{p^n} : K] by the tower law, so mm divides nn. For example, the subfields of 𝔽64=𝔽26\mathbb{F}_{64} = \mathbb{F}_{2^6} are 𝔽2\mathbb{F}_2, 𝔽4\mathbb{F}_4, 𝔽8\mathbb{F}_8 and 𝔽64\mathbb{F}_{64}. The subfield of order pmp^m is the fixed field of φm\varphi^m, where φ\varphi is the Frobenius automorphism.

Reading path

To understand this page from the ground up, read these 15 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 ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionField, definition, no evidence levelFielddefinitionGroup of units of a ring, definition, no evidence levelGroup of units of a r…definitionFinite field, definition, no evidence levelFinite fielddefinitionSubfield, definition, no evidence levelSubfielddefinitionEvery element of a finite field of order q satisfies x^q = x, statement, no evidence levelEvery element of a fi…statementField extension, definition, no evidence levelField extensiondefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionTower law, statement, no evidence levelTower lawstatementSubfields of a finite field, statement, no evidence levelSubfields 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.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Subring definition
  5. Unit of a ring definition
  6. Field definition
  7. Group of units of a ring definition
  8. Finite field definition
  9. Subfield definition
  10. Every element of a finite field of order q satisfies x^q = x statement
  11. Field extension definition
  12. Degree of a field extension definition
  13. Intermediate field definition
  14. Tower law statement
  15. Subfields of a finite field statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_field, Existence and uniqueness
About this page
Tag08YERZDM9X
URIm2://08YERZDM9X
Kindstatement, theorem
WritingW0 stub
MSC12E20
StatusThis copy of the site has no build data. Run m2 status 08YERZDM9X.
UsesFinite field, Tower law, Every element of a finite field of order q satisfies x^q = x, Subfield

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