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In mathematics,differential Galois theory is the field that studies extensions ofdifferential fields.

Whereas algebraicGalois theory studies extensions ofalgebraic fields, differential Galois theory studies extensions ofdifferential fields, i.e. fields that are equipped with aderivation,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 matrixLie groups, as compared with the finite groups often encountered in algebraic Galois theory.

Inmathematics, some types ofelementary functions cannot express theindefinite integrals of other elementary functions. A well-known example is${\displaystyle e^{-x^2}}$, whose indefinite integral is theerror function${\displaystyle \mathrm{erf}x}$, familiar instatistics. Other examples include thesinc function${\displaystyle \frac{\sin x}{x}}$ and${\displaystyle x^x}$.

It's important to note that the concept of elementary functions is merely conventional. If we redefine elementary functions to include the error function, then under this definition, the indefinite integral of${\displaystyle e^{-x^2}}$ would be considered an elementary function. However, no matter how many functions are added to the definition of elementary functions, there will always be functions whose indefinite integrals are not elementary.

Using the theory ofdifferential Galois theory , it is possible to determine which indefinite integrals of elementary functions cannot be expressed as elementary functions. Differential Galois theory is based on the framework ofGalois theory. While algebraic Galois theory studiesfield extensions offields, differential Galois theory studies extensions ofdifferential fields—fields with aderivationD.

Most of differential Galois theory is analogous to algebraic Galois theory. The significant difference in the structure is that the Galois group in differential Galois theory is analgebraic group, whereas in algebraic Galois theory, it is aprofinite group equipped with theKrull topology.

For any differential fieldF with derivationD, there exists a subfield called thefield of constants ofF, defined as:

Con(F) = {f ∈F |Df = 0}.

The field of constants contains the prime field ofF.

Given two differential fieldsF andG,G is called a simple differential extension ofF if^[1]and satisfies

∃s∈F;Dt =Ds/s,

thenG is called alogarithmic extension ofF.

This has the form of a logarithmic derivative. Intuitively,t can be thought of as the logarithm of some elements inF, corresponding to the usual chain rule.F does not necessarily have a uniquely defined logarithm. Various logarithmic extensions ofF can be considered. Similarly, anexponential extension satisfies

∃s∈F;Dt =tDs,

and adifferential extension satisfies

∃s∈F;Dt =s.

A differential extension or exponential extension becomes a Picard-Vessiot extension when the field has characteristic zero and the constant fields of the extended fields match.

Keeping the above caveat in mind, this element can be regarded as the exponential of an elements inF. Finally, if there is a finite sequence of intermediate fields fromF toG with Con(F) = Con(G), such that each extension in the sequence is either a finite algebraic extension, a logarithmic extension, or an exponential extension, thenG is called anelementary differential extension .

Consider the homogeneouslinear differential equation for${\displaystyle a_1,\cdots ,a_n\in F}$:

${\displaystyle D^{n}y+a_1{D}^{n-1}y+\cdots +a_{n-1}Dy+a_{n}y=0}$　…　(1).

There exist at mostn linearly independent solutions over the field of constants. An extensionG ofF is aPicard-Vessiot extension for the differential equation (1) ifG is generated by all solutions of (1) and satisfies Con(F) = Con(G).

An extensionG ofF is aLiouville extension if Con(F) = Con(G) is an algebraically closed field, and there exists an increasing chain of subfields

F =F₀ ⊂F₁ ⊂ … ⊂F_n =G

such that each extensionF_k+1 :F_k is either a finite algebraic extension, a differential extension, or an exponential extension. A Liouville extension of the rational function fieldC(x) consists of functions obtained by finite combinations of rational functions, exponential functions, roots of algebraic equations, and their indefinite integrals. Clearly, logarithmic functions, trigonometric functions, and their inverses are Liouville functions overC(x), and especially elementary differential extensions are Liouville extensions.

An example of a function that is contained in an elementary extension overC(x) but not in a Liouville extension is the indefinite integral of${\displaystyle e^{-x^2}}$.

For a differential fieldF, ifG is a separable algebraic extension ofF, the derivation ofF uniquely extends to a derivation ofG. Hence,G uniquely inherits the differential structure ofF.

SupposeF andG are differential fields satisfying Con(F) = Con(G), andG is an elementary differential extension ofF. Leta ∈F andy ∈G such thatDy =a (i.e.,G contains the indefinite integral ofa). Then there existc₁, …,c_n ∈ Con(F) andu₁, …,u_n,v ∈F such that

${\displaystyle a=c_1\frac{D{u}_1}{u_1}+\cdots +c_n\frac{D{u}_n}{u_n}+Dv}$

(Liouville's theorem). In other words, only functions whose indefinite integrals are elementary (i.e., at worst contained within the elementary differential extension ofF) have the form stated in the theorem. Intuitively, only elementary indefinite integrals can be expressed as the sum of a finite number of logarithms of simple functions.

IfG/F is a Picard-Vessiot extension, thenG being a Liouville extension ofF is equivalent to the differential Galois group having a solvable identity component.^[2] Furthermore,G being a Liouville extension ofF is equivalent toG being embeddable in some Liouville extension field ofF.

• The field of rational functions of one complex variableC(x) becomes a differential field when taking the usual differentiation with respect to the variablex as the derivation. The field of constants of this field is the complex number fieldC.

• ByLiouville's theorem mentioned above, iff(z) andg(z) are rational functions inz,f(z) is non-zero, andg(z) is non-constant, then${\displaystyle \int f(z)e^{g(z)}dz}$ is an elementary function if and only if there exists a rational functionh(z) such that${\displaystyle f(z)=h^{\prime }(z)+h(z)g^{\prime }(z)}$. The fact that theerror function and thesine integral (indefinite integral of thesinc function) cannot be expressed as elementary functions follows immediately from this property.

• In the case of the differential equation${\displaystyle y^{″}+y=0}$, the Galois group is the multiplicative group of complex numbers with absolute value 1, also known as thecircle group. This is an example of a solvable group, and indeed, the solutions to this differential equation are elementary functions (trigonometric functions in this case).

• The differential Galois group of theAiry equation,${\displaystyle y^{″}-xy=0}$, over the complex numbers is thespecial linear group of degree two, SL(2,C). This group is not solvable, indicating that its solutions cannot be expressed using elementary functions. Instead, the solutions are known as Airy functions.

Differential Galois theory has numerous applications in mathematics and physics. It is used, for instance, in determining whether a given differential equation can be solved by quadrature (integration). It also has applications in the study of dynamic systems, including the integrability of Hamiltonian systems in classical mechanics.

One significant application is the analysis of integrability conditions for differential equations, which has implications in the study of symmetries and conservation laws in physics.

• Picard–Vessiot theory

• Kolchin, E. R.,Differential Algebra and Algebraic Groups, Academic Press, 1973.
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• Juan J. Morales Ruiz :Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser, 1999, ISBN 978-3764360788 .

• Galois theory
• Differential algebra
• Differential equations
• Algebraic groups

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