Transcendental extension
A field extension that is not algebraic (cf. Extension of a field). An extension is transcendental if and only if the field
contains elements that are transcendental over
, that is, elements that are not roots of any non-zero polynomial with coefficients in
.
The elements of a set are called algebraically independent over
if for any finite set
and any non-zero polynomial
with coefficients in
,
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The elements of are transcendental over
. If
is a maximal set of algebraically independent elements over
, then
is called a transcendence basis of
over
. The cardinality of
is called the transcendence degree of
over
and is an invariant of the extension
. For a tower of fields
, the transcendence degree of
is equal to the sum of the transcendence degrees of
and
.
If all elements of a set are algebraically independent over
, then the extension
is called purely transcendental. In this case the field
is isomorphic to the field of rational functions in the set of variables
over
. Any field extension
can be represented as a tower of extensions
, where
is an algebraic and
is a purely transcendental extension. If
can be chosen so that
is a separable extension, then the extension
is called separably generated, and the transcendence basis of
over
is called a separating basis. If
is separably generated over
, then
is separable over
. In the case when the extension
is finitely generated, the converse holds as well. By definition, an extension
is separable if and only if any derivation (cf. Derivation in a ring) of
extends to
. Such an extension is uniquely determined for any derivation if and only if the extension
is algebraic.
References
[1] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |
[2] | N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Springer (1988) pp. Chapt. 4–6 (Translated from French) |
Comments
The Noether normalization lemma says that if is an integral domain that is finitely generated as a ring over a field
, then there exist
that are algebraically independent over
such that
is integral over
.
References
[a1] | P.M. Cohn, "Algebra" , 1–2 , Wiley (1989) pp. Vol. 2, 350; Vol. 3, 168ff |
Transcendental extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendental_extension&oldid=17184