In
mathematics, differential Galois theory studies the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
s of differential equations.
Overview
Whereas algebraic
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
studies extensions of
algebraic fields, differential Galois theory studies extensions of
differential field
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natu ...
s, i.e. fields that are equipped with a
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
, ''D''. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix
Lie groups, as compared with the finite groups often encountered in algebraic Galois theory.
See also
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Picard–Vessiot theory
In differential algebra, Picard–Vessiot theory is the study of the differential field extension generated by the solutions of a linear differential equation
In mathematics, a linear differential equation is a differential equation that is d ...
References
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*{{Citation , last1=van der Put , first1=Marius , last2=Singer , first2=Michael F. , title=Galois theory of linear differential equations , url=http://www4.ncsu.edu/~singer/ms_papers.html , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Grundlehren der Mathematischen Wissenschaften
undamental Principles of Mathematical Sciences, isbn=978-3-540-44228-8 , mr=1960772 , year=2003 , volume=328
Galois theory
Differential algebra
Differential equations
Algebraic groups