Differential Galois theory

Definition

For any differential field F with derivation D, there exists a subfield called the field of constants of F, defined as:

Con(F) = {f ∈ F | Df = 0}.

The field of constants contains the prime field of F.

Given two differential fields F and G, G is called a simple differential extension of F if for some element t, G = F<t> := F(t, Dt, D(Dt), …). If the simple extension G satisfies

∃s∈F; Dt = Ds/s,

then G is called a logarithmic extension of F.

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

∃s∈F; Dt = tDs,

and a differential extension satisfies

∃s∈F; Dt = s.

Keeping the above caveat in mind, this element can be regarded as the exponential of an element s in F. Finally, if there is a finite sequence of intermediate fields from F to G 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, then G is called an elementary differential extension .

Consider the homogeneous linear 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 most n linearly independent solutions over the field of constants. An extension G of F is a Picard-Vessiot extension for the differential equation (1) if G is generated by all solutions of (1) and satisfies Con(F) = Con(G). 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.

An extension G of F is a Liouville 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 extension F_k+1 : F_k is either a finite algebraic extension, a differential extension, or an exponential extension. A Liouville extension of the rational function field C(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 over C(x), and especially elementary differential extensions are Liouville extensions.

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

Basic properties

For a differential field F, if G is a separable algebraic extension of F, the derivation of F uniquely extends to a derivation of G. Hence, G uniquely inherits the differential structure of F.

Suppose F and G are differential fields satisfying Con(F) = Con(G), and G is an elementary differential extension of F. Let a ∈ F and y ∈ G such that Dy = a (i.e., G contains the indefinite integral of a). Then there exist c₁, …, c_n ∈ Con(F) and u₁, …, 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 of F) 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.

If G/F is a Picard-Vessiot extension, then G being a Liouville extension of F is equivalent to the differential Galois group having a solvable identity component. Furthermore, G being a Liouville extension of F is equivalent to G being embeddable in some Liouville extension field of F.

Examples

• The field of rational functions of one complex variable ${\displaystyle \mathbb{C}(x)}$ becomes a differential field when taking the usual differentiation with respect to the variable ${\displaystyle x}$ as the derivation. The field of constants of this field is the complex number field ${\displaystyle \mathbb{C}}$.

• By Liouville's theorem mentioned above, if ${\displaystyle f(z)}$ and ${\displaystyle g(z)}$ are rational functions in ${\displaystyle z}$, ${\displaystyle f(z)}$ is non-zero, and ${\displaystyle g(z)}$ is non-constant, then ${\displaystyle \int f(z)e^{g(z)}dz}$ is an elementary function if and only if there exists a rational function ${\displaystyle h(z)}$ such that ${\displaystyle f(z)=h^{\prime }(z)+h(z)g^{\prime }(z)}$. The fact that the error function and the sine integral (indefinite integral of the sinc 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 ${\displaystyle 1}$, also known as the circle 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 the Airy equation, ${\displaystyle y^{″}-xy=0}$, over the complex numbers is the special linear group of degree two, ${\displaystyle \mathrm{SL}(2,\mathbb{C})}$. This group is not solvable, indicating that its solutions cannot be expressed using elementary functions. Instead, the solutions are known as Airy functions.

Applications

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.