transcendence degree
التعريفات والمعاني
== English ==
=== Noun ===
transcendence degree (plural transcendence degrees)
(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
F
1
/
k
{\displaystyle F_{1}/k}
and
F
2
/
k
{\displaystyle F_{2}/k}
denote algebraic function fields of transcendence degree one.
==== Usage notes ====
A transcendence degree is said to be of a field extension (i.e.,
L
/
K
{\displaystyle L/K}
). More properly, it is the cardinality of a particular type of subset of the extension field
L
{\displaystyle L}
, although the context of the field extension is required to make sense of the definition.
Relatedly, a transcendence basis of
L
/
K
{\displaystyle L/K}
is a subset of
L
{\displaystyle L}
that is algebraically independent over
K
{\displaystyle K}
and such that
L
{\displaystyle L}
is an algebraic extension of
K
(
S
)
{\displaystyle K(S)}
(that is,
L
/
K
(
S
)
{\displaystyle L/K(S)}
is an algebraic extension).
It can be shown that every field extension
L
/
K
{\displaystyle L/K}
has a transcendence basis, whose cardinality, denoted
trdeg
K
L
{\displaystyle \operatorname {trdeg} _{K}L}
or
trdeg
(
L
/
K
)
{\displaystyle \operatorname {trdeg} (L/K)}
, is exactly the transcendence degree of
L
/
K
{\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