Differential Galois theory

integration in finite terms
quadrature of second order differential equation (Fuchsian differential equation)

일계 선형미분방정식
\(\frac{dy}{dx}+a(x)y=b(x)\)
\(y(x)e^{\int a(x)\,dx}=\int b(x)e^{\int a(x)\,dx} \,dx+C\)
\(y''-2xy'=0\)
\(y=\int e^{x^2}\, dx\)
note that the integral of an exponential naturally shows up in expression solutions

a pair \((F,\partial)\) such that
- \(\partial(a+b)=\partial a+\partial b\)
- \(\partial(ab)=(\partial a)b+a(\partial b)\)
\(C_F=\ker \partial\)

basic functions : basic elementary functions
allowed operatrions : compositions, arithmetic operations, differentiation, integration
examples
- an elliptic integral is representable by quadrature

it is allowed to take exponentials and logarithms to make a field extension
elementary element
difference between Liouville extension
- exponential+ integral <=> differentiation + exponential of integral
- in elementary extension, we are not allowed to get an integrated element

an element is said to be representable by a generalized quadrature
we can capture these properties using the concept of Liouville extension
to get a Liouville extension, we can adjoin
- integrals
- exponentials of integrals
- algebraic extension (generalized Liouville extension)
  - from these we can include the following operations
    - exponential
    - logarithm
For\(K_{i}=K_{i-1}(e_i)\) , one of the following condition holds
- \(e_i'\in K_{i-1}\), i.e. \(e_i=\int e_i'\in K_i\)
- \(e_{i}'/e_{i}\in K_{i-1}\) i.e. \((\log e_i)' \in K_{i-1}\)
- \(e_{i}\) is algebraic over \(K_{i-1}\)
remark on exponentiation
- Let \(a,a'\in F\). Is \(b=e^a\in K\) where K is a Liouville extension?
- \(b'=a' e^a=a'b\) implies \(a'=\frac{b'}{b}\in F\).
- the exponential of the integral of a' i.e. \(e^{\int a'}=e^a+c\) must be in the Liouville extension. So \(b=e^a\in K\).
remark on logarithm
- \(b=\log a\) is the integral of \(a'/a\in F\). So \(b\in K\)

a few result
- K/F is a Liouville extension iff the differential Galois group K over F is solvable.
- K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity

framework for linear differential equation
consider monic differential equations over a differential field F
\(\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0\), \(a_i\in F\)
made by including solutions of DE to the base field (e.g. rational function field)
this corresponds to the concept of the splitting fields(or Galois extensions)
we can define a Galois group for a linear differential equation.
examples
- algebraic extension
- adjoining an integral
- adjoining the exponential of an integral

theorem

If a Picard-Vessiot extension is a Liouville extension, then the Galois group of this extension is solvable.

theorem

A Fuchsian linear differential equation is solvable by quadratures if and only if the monodromy group of this equation is solvable.

Liouvillian First Integrals of Differential Equations
- Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
Elementary and Liouvillian solutions of linear differential equations
- M. F. Singer and J. H. Davenport, 1985

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