Math 2.0

statement theorem

A finite field of order q is the splitting field of x^q - x

Statement

Let FF be a finite field of order q=pnq = p^n, with pp prime. Then FF is a splitting field of xq−xx^q - x over its prime subfield Fp\mathbb{F}_p.

Remarks

The proof: xq−x=∏a∈F(x−a)x^q - x = \prod_{a \in F} (x - a), so xq−xx^q - x splits over FF, and its roots are all the elements of FF. So the roots generate FF. With the uniqueness of splitting fields, this gives the uniqueness of finite fields.

Explanations

L1 human

A finite field consists exactly of the roots of xq−xx^q - x. So it is the smallest field that contains all these roots.

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To read this node, read these 26 nodes in this order. The foundations come first, and the list ends with this node.

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 ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionGroup of units of a ring, definition, no evidence levelGroup of units of a r…definitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionThe characteristic of an integral domain is zero or prime, statement, no evidence levelThe characteristic of…statementFinite 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…statementSplit polynomial, definition, no evidence levelSplit polynomialdefinitionPrime subfield, definition, no evidence levelPrime subfielddefinitionField extension, definition, no evidence levelField extensiondefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionThe order of a finite field is a prime power, statement, no evidence levelThe order of a finite…statementField generated by adjoining elements, definition, no evidence levelField generated by ad…definitionSplitting field, definition, no evidence levelSplitting fielddefinitionA finite field of order q is the splitting field of x^q - x, statement, no evidence levelA finite field of ord…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. Characteristic of a ring definition
  4. Commutative ring definition
  5. Polynomial ring definition
  6. Subring definition
  7. Unit of a ring definition
  8. Zero divisor definition
  9. Group of units of a ring definition
  10. Polynomial evaluation definition
  11. Field definition
  12. Integral domain definition
  13. Root of a polynomial definition
  14. The characteristic of an integral domain is zero or prime statement
  15. Finite field definition
  16. Subfield definition
  17. Every element of a finite field of order q satisfies x^q = x statement
  18. Split polynomial definition
  19. Prime subfield definition
  20. Field extension definition
  21. Degree of a field extension definition
  22. Intermediate field definition
  23. The order of a finite field is a prime power statement
  24. Field generated by adjoining elements definition
  25. Splitting field definition
  26. A finite field of order q is the splitting field of x^q - x statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Finite_field, Existence and uniqueness

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