Field theory and polynomials Transcendental extensions Concept
Transcendence degree
Summary
The transcendence degree of a field extension is the cardinality of a transcendence basis. It counts the algebraically independent elements needed so that the rest is algebraic. It does not depend on the choice of basis. It is the field-theoretic notion of dimension: the function field of an irreducible variety has transcendence degree equal to the dimension of the variety.
Where this leads
11 pages build on this one.
- Dimension of an algebraic variety
- Model theory of algebraically closed fields
- Noether normalization lemma
- Steinitz's theorem on algebraically closed fields
- Function field
- Siegel's theorem on meromorphic function fields
- Algebraic dimension
- Hilbert's fourteenth problem
- Categorical theory
- Rational variety
- Tsen's theorem
Further reading
About this page
| Tag | D7TR0YWR9H |
|---|---|
| URI | m2://D7TR0YWR9H |
| Kind | concept |
| Writing | W0 stub |
| MSC | 12F20 |
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Used by
- Dimension of an algebraic variety
- Noether normalization lemma
- Steinitz's theorem on algebraically closed fields
- Function field
- Siegel's theorem on meromorphic function fields
- Algebraic dimension
- Hilbert's fourteenth problem
- Rational variety
- Tsen's theorem