Algebraic independence: Difference between revisions
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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. | 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. | ||
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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 | 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== | ||
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==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 12:01, 8 July 2024
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
- The singleton set {s} is algebraically independent if and only if s is transcendental over F.
References
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 355-357. ISBN 0-201-55540-9.