Algebraic independence: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(subpages)
mNo edit summary
 
(One intermediate revision by one other user not shown)
Line 11: Line 11:
An algebraically independent subset of ''E'' of maximal [[cardinality]] is a '''transcendence basis''' for ''E''/''F'', and this cardinality is the '''transcendence degree''' or '''transcendence dimension''' of ''E'' over ''F''.
An algebraically independent subset of ''E'' of maximal [[cardinality]] is a '''transcendence basis''' for ''E''/''F'', and this cardinality is the '''transcendence degree''' or '''transcendence dimension''' of ''E'' over ''F''.


Algebraic independence has the ''[[exchange property]]'': if ''G'' is a set such that ''E'' is algebraic over ''F''(''G''), and ''I'' is a subset of ''G'' which is algebraically independent, then there is a subset ''B'' of ''G'' with <math>I \subseteq B \subseteq G</math> which is a transcendence basis.  The algebraically independent subsets thus form the independent sets of a [[matroid]].  
Algebraic independence has the ''[[exchange property]]'': if ''G'' is a set such that ''E'' is algebraic over ''F''(''G''), and ''I'' is a subset of ''G'' which is algebraically independent, then there is a subset ''B'' of ''G'' with <math>I \subseteq B \subseteq G</math> which is a transcendence basis.  The algebraically independent subsets thus form an [[independence structure]].


==Examples==
==Examples==
Line 17: Line 17:


==References==
==References==
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=355-357 }}
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=355-357 }}[[Category:Suggestion Bot Tag]]

Latest revision as of 11:01, 8 July 2024

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In algebra, algebraic independence is a property of a set of elements of an extension field E/F, that they satisfy no non-trivial algebraic relation.

Formally, a subset S of E is algebraically independent over F if any polynomial with coefficients in F, say f(X1,...,Xn), such that f(s1,...,sn)=0 where the si are distinct elements of S, must be zero as a polynomial.

If there is a non-zero polynomial f such that f(s1,...,sn)=0, then the si are said to be algebraically dependent.

Any subset of an algebraically independent set is algebraically independent.

An algebraically independent subset of E of maximal cardinality is a transcendence basis for E/F, and this cardinality is the transcendence degree or transcendence dimension of E over F.

Algebraic independence has the exchange property: if G is a set such that E is algebraic over F(G), and I is a subset of G which is algebraically independent, then there is a subset B of G with which is a transcendence basis. The algebraically independent subsets thus form an independence structure.

Examples

References