Math 2.0

Field theory and polynomials Definition

Purely transcendental extension

Builds on Field extensionAlgebraic independenceField of rational functionsLüroth's theorem

Definition

Let L/KL/K be a field extension. The extension L/KL/K is purely transcendental if L=K(S)L = K(S) for a subset S⊆LS \subseteq L that is algebraically independent over KK.

Remarks

Then LL is isomorphic over KK to the field of rational functions K(xs:s∈S)K(x_s : s \in S), by sending xsx_s to ss. Example: ℚ(π)/ℚ\Q(\pi)/\Q is purely transcendental, with S={π}S = \set{\pi}. Non-example: ℚ(2,t)/ℚ\Q(\sqrt 2, t)/\Q with tt transcendental is not purely transcendental. Indeed, KK is algebraically closed in K(S)K(S) for independent SS, but 2\sqrt 2 is algebraic over ℚ\Q and not in ℚ\Q. By Lüroth's theorem, every field strictly between KK and K(t)K(t) is again purely transcendental over KK.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Algebraic independence, concept, no evidence levelAlgebraic independenceconceptField of rational functions, concept, no evidence levelField of rational fun…conceptGroup, concept, no evidence levelGroupconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionPurely transcendental extension, concept, no evidence levelPurely transcendental…conceptSequence, definition, no evidence levelSequencedefinitionRing, definition, no evidence levelRingdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionField, definition, no evidence levelFielddefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSequential continuity, definition, no evidence levelSequential continuitydefinitionSubfield, definition, no evidence levelSubfielddefinitionContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementField extension, definition, no evidence levelField extensiondefinitionLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionProduct of continuous functions is continuous, statement, no evidence levelProduct of continuous…statementLimit of a quotient of sequences, statement, no evidence levelLimit of a quotient o…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …statementQuotient of continuous functions is continuous, statement, no evidence levelQuotient of continuou…statementRational function, definition, no evidence levelRational functiondefinitionLüroth's theorem, statement, no evidence levelLüroth's theoremstatementPurely transcendental extension, definition, no evidence levelPurely transcendental…definition
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. Algebraic independence concept
  2. Field of rational functions concept
  3. Group concept
  4. Limit point of a set definition
  5. Purely transcendental extension concept
  6. Sequence definition
  7. Ring definition
  8. Limit of a function definition
  9. Real sequence definition
  10. Commutative ring definition
  11. Subring definition
  12. Unit of a ring definition
  13. Continuity at a point definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Field definition
  17. Continuous function definition
  18. Epsilon-delta definition of continuity statement
  19. Convergent sequence definition
  20. Epsilon-N definition of the limit of a sequence statement
  21. Sequential continuity definition
  22. Subfield definition
  23. Continuity is equivalent to sequential continuity statement
  24. Convergent sequences are bounded statement
  25. Field extension definition
  26. Limit of a product of sequences statement
  27. Degree of a field extension definition
  28. Product of continuous functions is continuous statement
  29. Limit of a quotient of sequences statement
  30. Polynomial functions are continuous statement
  31. Quotient of continuous functions is continuous statement
  32. Rational function definition
  33. Lüroth's theorem statement
  34. Purely transcendental extension definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Transcendental_extension
About this page
Tag7DKKTMPE6V
URIm2://7DKKTMPE6V
Kinddefinition
WritingW0 stub
MSC12F20
DefinesPurely transcendental extension
UsesField extension, Algebraic independence, Field of rational functions, Lüroth's theorem
Changed2 November 2026

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