View source for Transcendence degree

==English==
{{wikipedia}}

===Noun===
{{en-noun}}

# {{lb|en|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'', {{w|Lecture Notes in Computer Science|LNCS}} 3076, [https://books.google.com.au/books?id=LpYCsnunhOUC&pg=PA263&dq=%22transcendence+degree%22%7C%22transcendence+degrees%22&hl=en&sa=X&ved=0ahUKEwjByOTDh_ffAhXTHzQIHZoLAaYQ6AEIjAEwEg#v=onepage&q=%22transcendence%20degree%22%7C%22transcendence%20degrees%22&f=false page 263],
#*: Let <math>F_1/k</math> and <math>F_2/k</math> denote algebraic function fields of '''transcendence degree''' one.
#* {{quote-book|en|year=2007|author=w:Anthony W. Knapp|title=Advanced Algebra|publisher=Springer (Birkhäuser)|pageurl=https://books.google.com.au/books?id=25JfJAgqC8sC&pg=PA422&dq=%22transcendence+degree%22%7C%22transcendence+degrees%22&hl=en&sa=X&ved=0ahUKEwjByOTDh_ffAhXTHzQIHZoLAaYQ6AEIMDAB#v=onepage&q=%22transcendence%20degree%22%7C%22transcendence%20degrees%22&f=false|page=422
|passage='''Lemma 7.19''' Suppose that <math>L</math> is a field extension{{sic|extension field}} of '''transcendence degree''' <math>r</math> over a field <math>K</math> and that <math>L</math> is not separably generated over <math>K</math>. If <math>x_1,\dots,x_n</math> are elements of <math>L</math> such that <math>L=K(x_1,\dots,x_n)</math>, then for a suitable relabeling of the <math>x_i</math>'s, the subfield <math>K(x_1,\dots,x_{r+1})</math> of <math>L</math> is of '''transcendence degree''' <math>r</math> and is not separably generated over <math>K</math>.}}
#* {{quote-book|en|year=2008|author=w:Bernd Sturmfels|title=Algorithms in Invariant Theory|edition=2nd|pageurl=https://books.google.com.au/books?id=3lF0cyOHwgQC&pg=PA24&dq=%22transcendence+degree%22%7C%22transcendence+degrees%22&hl=en&sa=X&ved=0ahUKEwjByOTDh_ffAhXTHzQIHZoLAaYQ6AEIVDAI#v=onepage&q=%22transcendence%20degree%22%7C%22transcendence%20degrees%22&f=false|page=24|publisher=Springer
|passage='''Proposition 2.1.1''' Every finite matrix group <math>\Gamma\subset\operatorname{GL}(\C^n)</math> has <math>n</math> algebraically independent invariants, i.e., the ring <math>\C[x]^\Gamma</math> has '''transcendence degree''' <math>n</math> over <math>\C</math>.}}

====Usage notes====
* A transcendence degree is said to be '''''of''''' a field extension (i.e., <math>L/K</math>). More properly, it is the cardinality of a particular type of subset of the extension field <math>L</math>, although the context of the field extension is required to make sense of the definition.
* Relatedly, a {{m|en|transcendence basis}} of <math>L/K</math> is a subset of <math>L</math> that is [[algebraically independent]] over <math>K</math> and such that <math>L</math> is an [[algebraic extension]] of <math>K(S)</math> (that is, <math>L/K(S)</math> is an algebraic extension).
** It can be shown that every field extension <math>L/K</math> has a transcendence basis, whose cardinality, denoted <math>\operatorname{trdeg}_K L</math> or <math>\operatorname{trdeg}(L/K)</math>, is exactly the transcendence degree of <math>L/K</math>.

====Synonyms====
* {{sense|cardinality of largest algebraically independent subset of a given extension field}} {{l|en|transcendental degree}}

====Related terms====
* {{l|en|transcendence}}
* {{l|en|transcendence basis}}

====Translations====
{{trans-top|cardinality of largest algebraically independent subset of a given extension field}}
{{trans-bottom}}

===See also===
* {{l|en|algebraically independent}}

===Further reading===
* {{pedia|Algebraic independence}}
* [https://www.encyclopediaofmath.org/index.php/Algebraic_independence Algebraic independence] on {{w|Encyclopedia of Mathematics}}
* [http://mathworld.wolfram.com/TranscendenceDegree.html  Transcendence Degree] on {{w|MathWorld|Wolfram MathWorld}}