transcendence degree

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transcendence degree

Wikipedia

Noun

transcendence degree (plural transcendence degrees)

1. (algebra, field theory, of a field extension) 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, LNCS 3076, page 263,

     Let ${\displaystyle F_{1}/k}$ and ${\displaystyle F_{2}/k}$ denote algebraic function fields of transcendence degree one.

   • 2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422:

     Lemma 7.19 Suppose that ${\displaystyle L}$ is a field extension of transcendence degree ${\displaystyle r}$ over a field ${\displaystyle K}$ and that ${\displaystyle L}$ is not separably generated over ${\displaystyle K}$. If ${\displaystyle x_{1},\dots ,x_{n}}$ are elements of ${\displaystyle L}$ such that ${\displaystyle L=K(x_{1},\dots ,x_{n})}$, then for a suitable relabeling of the ${\displaystyle x_{i}}$'s, the subfield ${\displaystyle K(x_{1},\dots ,x_{r+1})}$ of ${\displaystyle L}$ is of transcendence degree ${\displaystyle r}$ and is not separably generated over ${\displaystyle K}$.

   • 2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24:

     Proposition 2.1.1 Every finite matrix group ${\displaystyle \Gamma \subset \operatorname {GL} (\mathbb {C} ^{n})}$ has ${\displaystyle n}$ algebraically independent invariants, i.e., the ring ${\displaystyle \mathbb {C} ^{\Gamma }}$ has transcendence degree ${\displaystyle n}$ over ${\displaystyle \mathbb {C} }$.

Usage notes

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

Synonyms

• (cardinality of largest algebraically independent subset of a given extension field): transcendental degree

Related terms

• transcendence
• transcendence basis

Translations

See also

• algebraically independent

Further reading

• Algebraic independence on Wikipedia.Wikipedia
• Algebraic independence on Encyclopedia of Mathematics
• Transcendence Degree on Wolfram MathWorld