Math 2.0

Field theory and polynomials Definition

Transcendental extension

Builds on Field extensionAlgebraic extensionTranscendental elementTranscendental number

Definition

Let L/KL/K be a field extension. The extension L/KL/K is transcendental if it is not algebraic, that is, if some element of LL is transcendental over KK.

Remarks

Example: ℝ/ℚ\R/\Q is transcendental, because π\pi is a transcendental number. Example: for a field KK, the extension K(t)/KK(t)/K of rational functions is transcendental, since g(t)≠0g(t) \ne 0 for every nonzero g∈K[x]g \in K[x]. Non-example: ℚ(2)/ℚ\Q(\sqrt 2)/\Q is not transcendental, because it is finite and hence algebraic. Every field extension L/KL/K is the composite of a purely transcendental step K⊆K(B)K \subseteq K(B) and an algebraic step K(B)⊆LK(B) \subseteq L, where BB is a transcendence basis.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Algebraic number, definition, no evidence levelAlgebraic numberdefinitionGroup, concept, no evidence levelGroupconceptTranscendental number, definition, no evidence levelTranscendental numberdefinitionRing, 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 ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionField extension, definition, no evidence levelField extensiondefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionTranscendental element, definition, no evidence levelTranscendental elementdefinitionTranscendental extension, definition, no evidence levelTranscendental extens…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 number definition
  2. Group concept
  3. Transcendental number definition
  4. Ring definition
  5. Commutative ring definition
  6. Polynomial ring definition
  7. Subring definition
  8. Unit of a ring definition
  9. Polynomial evaluation definition
  10. Field definition
  11. Root of a polynomial definition
  12. Subfield definition
  13. Field extension definition
  14. Algebraic element definition
  15. Algebraic extension definition
  16. Transcendental element definition
  17. Transcendental extension definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Transcendental_extension
About this page
TagHMF8G4EBDX
URIm2://HMF8G4EBDX
Kinddefinition
WritingW0 stub
MSC12F20
DefinesTranscendental extension
UsesField extension, Algebraic extension, Transcendental element, Transcendental number
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.