Finite extension: Difference between revisions

Latest revision as of 16:22, 10 May 2009

This article defines a field extension property: a property that can be evaluated to true/false for any field extension.
View a complete list of field extension properties|View a complete list of field properties

Definition

Definition with symbols

Suppose $L$ is a field extension of a field $K$ (i.e., $K$ is a subfield of $L$ ). We say that $L$ is a finite extension of $K$ if it satisfies the following equivalent conditions:

The degree of the extension, denoted $[L:K]$ , is finite, where the degree is defined as the dimension of $L$ as a $K$ -vector space.
$L$ is generated over $K$ by a finite collection of elements, each of which is algebraic over $K$ .
$L$ is an algebraic extension of $K$ that is also finitely generated.
$L$ is a simple extension of $K$ that is also algebraic.
There is an irreducible polynomial $p(x)\in K[x]$ and an isomorphism $L\cong K[x]/(p(x))$ , such that the isomorphism restricts to the identity on $K$ .

Relation with other properties

Stronger properties

Weaker properties

@@ Line 5: / Line 5: @@
 ===Definition with symbols===
-Suppose <math>L</math> is a field extension of a field <math>K</math> (i.e., <math>K</math> is a subfield of <math>L</math>). We say that <math>L</math> is a '''finite extension''' of <math>K</math> if the [[defining ingredient::degree of a field extension|degree of the extension]], denoted <math>[L:K]</math>, is finite, where the degree is defined as the dimension of <math>L</math> as a <math>K</math>-vector space.
+Suppose <math>L</math> is a field extension of a field <math>K</math> (i.e., <math>K</math> is a subfield of <math>L</math>). We say that <math>L</math> is a '''finite extension''' of <math>K</math> if it satisfies the following equivalent conditions:
+# The [[defining ingredient::degree of a field extension|degree of the extension]], denoted <math>[L:K]</math>, is finite, where the degree is defined as the dimension of <math>L</math> as a <math>K</math>-vector space.
+# <math>L</math> is generated over <math>K</math> by a finite collection of elements, each of which is [[algebraic element|algebraic]] over <math>K</math>.
+# <math>L</math> is an [[defining ingredient::algebraic extension]] of <math>K</math> that is also [[defining ingredient::finitely generated extension|finitely generated]].
+# <math>L</math> is a simple extension of <math>K</math> that is also algebraic.
+# There is an irreducible polynomial <math>p(x) \in K[x]</math> and an isomorphism <math>L \cong K[x]/(p(x))</math>, such that the isomorphism restricts to the identity on <math>K</math>.
+==Relation with other properties==
+===Stronger properties===
+* [[Weaker than::Finite normal extension]]
+* [[Weaker than::Finite separable extension]]
+* [[Weaker than::Finite Galois extension]]
+===Weaker properties===
+* [[Stronger than::Algebraic extension]]
+* [[Stronger than::Finitely generated extension]]