In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 fields, 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 group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, as compared with the finite groups often encountered in algebraic Galois theory.
See also
*
Picard–Vessiot theory
References
*
*
*
*
*
*{{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