Math 2.0

Field theory and polynomials Field extensions Theorem

Primitive element theorem

Builds on Separable extensionFinite extensionSimple extension

Statement

Let L/KL/K be a finite separable extension. Then L/KL/K is a simple extension: there is θ∈L\theta \in L with L=K(θ)L = K(\theta).

Remarks

Every finite extension of a field of characteristic 00 is separable, so it is simple. For example, ℚ(2,3)=ℚ(2+3)\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3}). For an infinite field KK, the proof shows that θ=a+cb\theta = a + c b works for all but finitely many c∈Kc \in K. For finite fields, a primitive element of LL generates LL over KK. Without separability the statement can fail: there are finite extensions of degree p2p^2 in characteristic pp with no primitive element.

History

Évariste Galois sketched a proof for splitting fields over the rationals in his First Memoir of 1831, published in 1846. Ernst Steinitz proved the theorem in its modern form in 1910. Emil Artin later developed Galois theory without primitive elements.

Reading path

To understand this page from the ground up, read these 26 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 ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionFormal derivative, definition, no evidence levelFormal derivativedefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionSeparable polynomial, definition, no evidence levelSeparable polynomialdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionFinite extension, definition, no evidence levelFinite extensiondefinitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionField generated by adjoining elements, definition, no evidence levelField generated by ad…definitionSeparable extension, definition, no evidence levelSeparable extensiondefinitionSimple extension, definition, no evidence levelSimple extensiondefinitionPrimitive element theorem, statement, no evidence levelPrimitive element the…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. Polynomial ring definition
  5. Subring definition
  6. Unit of a ring definition
  7. Degree of a polynomial definition
  8. Formal derivative definition
  9. Polynomial evaluation definition
  10. Field definition
  11. Leading coefficient definition
  12. Separable polynomial definition
  13. Root of a polynomial definition
  14. Subfield definition
  15. Monic polynomial definition
  16. Field extension definition
  17. Degree of a field extension definition
  18. Algebraic element definition
  19. Intermediate field definition
  20. Finite extension definition
  21. Minimal polynomial of an algebraic element definition
  22. Algebraic extension definition
  23. Field generated by adjoining elements definition
  24. Separable extension definition
  25. Simple extension definition
  26. Primitive element theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Primitive_element_theorem, Theorem statement
  2. Web https://en.wikipedia.org/wiki/Primitive_element_theorem, History
About this page
Tag0SVX98HYV1
URIm2://0SVX98HYV1
Kindstatement, theorem
WritingW1 stated
MSC12F10
StatusThis copy of the site has no build data. Run m2 status 0SVX98HYV1.
UsesSeparable extension, Finite extension, Simple extension
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