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
field extension is made by including solutions of DE to the base field (e.g. rational function field)
consider monic differential equations over a differential field F
\(\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0\), \(a_i\in F\)
\((E,\partial_E)\supseteq (F,\partial_F)\) is a Picard-Vessiot extension for \(\mathcal{L}\) if
- E/F is generated by n linear independent solution to \(\mathcal{L}\), i.e. adjoining basis of \(V=\mathcal{L}^{-1}(0)\) to F
- \(C_E=C_F\), \(\partial_E\mid_F=\partial_F\)
this corresponds to the concept of the splitting fields(or Galois extensions)
examples
- algebraic extension
- adjoining an integral
- adjoining the exponential of an integral
we can define a Galois group for a linear differential equation
\(\operatorname{Gal}(E/F)=\{\sigma\in\operatorname{Aut}E|\partial(\sigma(n))=\sigma(\partial a), \sigma\mid _F=\operatorname{id} \}\)
- the action of an element of the Galois group is determined by its action on a basis of V

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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