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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
and
denote algebraic function fields of transcendence degree one.
2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422:Lemma 7.19 Suppose that
is a field extension of transcendence degree
over a field
and that
is not separably generated over
. If
are elements of
such that
, then for a suitable relabeling of the
's, the subfield
of
is of transcendence degree
and is not separably generated over
.
2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24:Proposition 2.1.1 Every finite matrix group
has
algebraically independent invariants, i.e., the ring
has transcendence degree
over
.
Usage notes
- A transcendence degree is said to be of a field extension (i.e.,
). More properly, it is the cardinality of a particular type of subset of the extension field
, although the context of the field extension is required to make sense of the definition.
- Relatedly, a transcendence basis of
is a subset of
that is algebraically independent over
and such that
is an algebraic extension of
(that is,
is an algebraic extension).
- It can be shown that every field extension
has a transcendence basis, whose cardinality, denoted
or
, is exactly the transcendence degree of
.
Synonyms
- (cardinality of largest algebraically independent subset of a given extension field): transcendental degree
Translations
cardinality of largest algebraically independent subset of a given extension field
See also
Further reading