Transcendence degree

English

Noun

1. Template:Lb Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K.
   • 2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, Template:W 3076, page 263,

     Let $F_1/k$ and $F_2/k$ denote algebraic function fields of transcendence degree one.

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Usage notes

• A transcendence degree is said to be of a field extension (i.e., $L/K$). More properly, it is the cardinality of a particular type of subset of the extension field $L$, although the context of the field extension is required to make sense of the definition.
• Relatedly, a Template:M of $L/K$ is a subset of $L$ that is algebraically independent over $K$ and such that $L$ is an algebraic extension of $K(S)$ (that is, $L/K(S)$ is an algebraic extension).
  • It can be shown that every field extension $L/K$ has a transcendence basis, whose cardinality, denoted ${\mathrm{trdeg}}_{K}L$ or $\mathrm{trdeg}(L/K)$, is exactly the transcendence degree of $L/K$.

Synonyms

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Related terms

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See also

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Further reading

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• Algebraic independence on Template:W
• Transcendence Degree on Template:W