Math 2.0

Field theory and polynomials Definition

Transcendence basis

Builds on Field extensionAlgebraic independenceAlgebraic extensionField of rational functions

Definition

Let L/KL/K be a field extension. A subset B⊆LB \subseteq L is a transcendence basis of L/KL/K if:

  1. BB is algebraically independent over KK;
  2. LL is an algebraic extension of the subfield K(B)K(B) generated by KK and BB.

Remarks

Equivalently, BB is a maximal algebraically independent subset of LL. Example: in the field of rational functions K(x1,…,xn)K(x_1, \dots, x_n), the set {x1,…,xn}\set{x_1, \dots, x_n} is a transcendence basis over KK, and so is {x12,x2,…,xn}\set{x_1^2, x_2, \dots, x_n}. Non-example: {x1}\set{x_1} is not a transcendence basis of K(x1,x2)/KK(x_1, x_2)/K, because x2x_2 is not algebraic over K(x1)K(x_1), so clause 2 fails. For an algebraic extension, the empty set is the unique transcendence basis.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Algebraic independence, concept, no evidence levelAlgebraic independenceconceptField of rational functions, concept, no evidence levelField of rational fun…conceptGroup, 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 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 extensiondefinitionTranscendence basis, definition, no evidence levelTranscendence basisdefinition
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. 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. Transcendence basis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Transcendental_extension
About this page
TagCYCJ46W6P6
URIm2://CYCJ46W6P6
Kinddefinition
WritingW0 stub
MSC12F20
DefinesTranscendence basis
UsesField extension, Algebraic independence, Algebraic extension, Field of rational functions
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • 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.