extension field
التعريفات والمعاني
== English ==
=== Noun ===
extension field (plural extension fields)
(algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.
==== Usage notes ====
Not to be confused with field extension, which refers to the construction
L
/
K
{\displaystyle L/K}
The extension field
L
{\displaystyle L}
constitutes a vector space over
K
{\displaystyle K}
(i.e., a
K
{\displaystyle K}
-vector space).
A minimal set
B
{\displaystyle B}
comprising one element of
K
{\displaystyle K}
plus additional elements not in
K
{\displaystyle K}
which together generate
L
{\displaystyle L}
is called a basis.
The dimension of the vector space (aka the degree of the extension), is denoted
[
L
:
K
]
{\displaystyle [L:K]}
and is equal to the cardinality of
B
{\displaystyle B}
.
In the case
L
=
K
{\displaystyle L=K}
,
L
{\displaystyle L}
is called the trivial extension and can be regarded as a vector space of dimension 1.
An extension field of degree 2 (respectively, 3) may be called a quadratic extension (respectively, cubic extension).
A field
F
{\displaystyle F}
which is both a subfield of
L
{\displaystyle L}
and an extension field of
K
{\displaystyle K}
may be called an intermediate field, intermediate extension or subextension of the field extension
L
/
K
{\displaystyle L/K}
.
==== Synonyms ====
(field that contains a subfield): extension (where the base field is given)
==== Hyponyms ====
number field
splitting field
==== Related terms ====
field extension
==== Translations ====
=== Further reading ===
Field extension on Wikipedia.Wikipedia
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 ===
field extension