Abel's theorem in problems and solutions - School of Mathematics

Solvability of Equations 245
corresponds a linear transform of the space into itself. The totality
of these linear transforms of the space forms a group which is called
the monodromy group of the equation (A. 14). So the monodromy group
of an equation is the group of the linear transforms of the solutions which
correspond to different turns around the singular points. The monodromy
group of an equation characterizes the property of its solutions being
multi- valued.
Near a non-singular point there are linearly independent solutions,
of the equation (A.14). In this neighbourhood one can
consider the field of functions that is obtained by adding to
the field of rational functions all solutions and all their derivatives.
Every transformation of the monodromy group of the space of solutions
can be continued to an automorphism of the entire field
Indeed, with functions along the curve every element of the
field can be analytically continued. This continuation gives
the required automorphism, because during the continuation the arithmetic
operations and the differentiation are preserved, and the rational
functions come back to their initial values because of their uniqueness.
In this way the monodromy group of an equation is contained in its
Galois group.
The field of the invariants of the monodromy group is a subfield of
consisting of the single-valued functions. Differently from
the algebraic case, for differential equations the field of invariants under
the action of the monodromy group can be bigger than the field of rational
functions.
For example, for the differential equation (A.14), in which all the coefficients
are polynomials, all solutions are single-valued. But of
course the solutions of such equations are not always polynomials. The
reason is that here the solutions of differential equations may grow exponentially
in approaching the singular points. One knows an extension
of the class of linear differential equations for which there are no similar
complications, i.e., for which the solutions, whilst approaching the singular
points, grow at most as some power. Differential equations which
possess this property are called equations of Fuchs’ type.
For differential equations of Fuchs’ type the Frobenius theorem holds.
THEOREM 1. For the differential equations of Fuchs’ type the subfield
of the differential field that consists of single-valued
functions coincides with the field of rational functions.
According to the differential Galois theory, from the Frobenius the-