field extension
التعريفات والمعاني
== English ==
=== Noun ===
field extension (plural field extensions)
(algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.
1998, David Goss, Basic Structures of Function Field Arithmetic, Springer, Corrected 2nd Printing, page 283,
Note that the extension of L obtained by adjoining all division points of
ψ
{\displaystyle \psi }
includes at most a finite constant field extension.
2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2bd Edition, page 530,
A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.
==== Usage notes ====
Related terminology:
L
{\displaystyle L}
may be said to be an extension field (or simply an extension) of
K
{\displaystyle K}
.
If a field
F
{\displaystyle F}
exists which is a subfield of
L
{\displaystyle L}
and of which
K
{\displaystyle K}
is a subfield, then we may call
F
{\displaystyle F}
an intermediate field (of
L
/
K
{\displaystyle L/K}
), or an intermediate extension or subextension (of
K
{\displaystyle K}
, or perhaps of
L
/
K
{\displaystyle L/K}
).
The field
L
{\displaystyle L}
is a
K
{\displaystyle K}
-vector space. Its dimension is called the degree of the extension, denoted
[
L
:
K
]
{\displaystyle [L:K]}
.
The construction
L
/
L
{\displaystyle L/L}
is called the trivial extension.
Field extensions are fundamental in algebraic number theory and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
==== Hyponyms ====
algebraic extension
transcendental extension
simple extension
trivial extension
==== Meronyms ====
base field
extension field
==== Related terms ====
extension
extension field
==== Translations ====
=== Further reading ===
Field theory on Wikipedia.Wikipedia
Glossary of field theory on Wikipedia.Wikipedia
Tower of fields on Wikipedia.Wikipedia
Primary extension on Wikipedia.Wikipedia
Regular extension on Wikipedia.Wikipedia
Extension Field on Wolfram MathWorld
Extension of a field on Encyclopedia of Mathematics
=== Anagrams ===
extension field