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

The complex numbers 75
in such functions. In fact, the final aim of our study is the proof of the
Abel theorem, according to which a function, expressing the roots of the
general equation of fifth degree, cannot be represented by radicals. But
this function is multi-valued, because an equation of fifth degree has, in
general, for given coefficients, five roots. Also the functions which are
represented by radicals are multi-valued.
The principal idea of the demonstration of the Abel theorem is the
following. We put into correspondence with a multi-valued function of a
complex variable a certain group, the so called monodromy group.
The monodromy group of the function expressing the roots of the
general equation of fifth degree in terms of a parameter cannot coincide
with any monodromy group of functions representable by radicals, and
therefore this function cannot be represented by radicals.
In order to introduce the notion of the monodromy group we consider
first another notion very important in the theory of functions of one
complex variable — the notion of the Riemann 12 surface of a function.
We begin by the construction of the Riemann surface for the simplest
example of a multi-valued function, the function
We already know that the function takes the single value
for and two values for all values (cf., 229). Moreover,
if is one of the values of then the other value of is
277. Find all values of: a) b) c) d) (here
is the positive value of the root).
Let us cut the plane along the negative side of the real axis from 0
to and for every not belonging to the cut let us choose the value
which lies on the right half plane. In this way we obtain a
continuous single-valued function over the whole plane, except the cut.
This function, which we denote by defines a continuous and singlevalued
mapping of the plane, except the cut, on the right half plane
(Figure 24).
REMARK. If we choose arg in such a way that then
for the function we obtain (cf., 229). Therefore
under the mapping the plane shrinks like a fan whose radii are
shortened as its opening angle is halved.
If we now choose, for every not lying on the cut, the value of
which lies on the left half plane we obtain another function, still single-
12 B. Riemann (1826–1866), German mathematician.