Differential Galois theory
Encyclopedia
In mathematics
, differential Galois theory studies the Galois group
s of differential equations.
Whereas algebraic Galois theory
studies extensions of algebraic fields
, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation
, 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. The problem of finding which integral
s of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of polynomial equations by radicals
in algebraic Galois theory, and is solved by Picard–Vessiot theory
.
called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension
of F (i.e. G = F(t) for some transcendental t) such that
This has the form of a logarithmic derivative
. Intuitively, one may think of t as the logarithm
of some element s of F, in which case, this condition is analogous to the ordinary chain rule
. But it must be remembered that F is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to F. Similarly, an exponential extension is a simple transcendental extension which satisfies
With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called an elementary differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, differential Galois theory studies the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s of differential equations.
Whereas algebraic Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
studies extensions of algebraic fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation
Derivation (abstract algebra)
In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...
, 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. The problem of finding which integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of polynomial equations by radicals
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
in algebraic Galois theory, and is solved by Picard–Vessiot theory
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, using the differential Galois group of the field extension. A major goal is to describe when the differential equation can be solved by...
.
Definitions
For any differential field F, there is a subfield- Con(F) = {f in F | Df = 0},
called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of F (i.e. G = F(t) for some transcendental t) such that
- Dt = Ds/s for some s in F.
This has the form of a logarithmic derivative
Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
. Intuitively, one may think of t as the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
of some element s of F, in which case, this condition is analogous to the ordinary chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
. But it must be remembered that F is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to F. Similarly, an exponential extension is a simple transcendental extension which satisfies
- Dt = tDs.
With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called an elementary differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.
See also
- Liouville's theorem (differential algebra)Liouville's theorem (differential algebra)In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions....
- Risch algorithmRisch algorithmThe Risch algorithm, named after Robert Henry Risch, is an algorithm for the calculus operation of indefinite integration . The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational...