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

2
After some transformations we obtain
In high school one considers only the case Indeed, if
then one says that Eq. (1) cannot be satisfied and that Eq.
(2) has no real roots. In order to avoid these exclusions, in what follows
we shall not restrict ourselves to algebraic equations over the field of the
real numbers, but we will consider them over the wider field of complex
numbers.
We shall examine complex numbers in greater detail (together with
their definition) in Chapter 2. In the meantime it is sufficient for the
reader to know, or to accept as true, the following propositions about the
complex numbers:
1.
2.
3.
the set of complex numbers is an extension of the set of real numbers,
i.e., the real numbers are contained in the complex numbers,
just as, for example, the integer numbers are contained in the real
numbers;
the complex numbers may be added, subtracted, multiplied, divided,
raised to a natural power; moreover, all these operations
possess all the basic properties of the corresponding operations on
the real numbers;
if is a complex number different from zero, and is a natural
number, then there exist exactly roots of degree of i.e.,
complex numbers such that For we have
If and are square roots of the number then
In the following we shall be interested not only in complex roots of
equations as well as in the real ones, but also we will consider arbitrary
complex numbers as coefficients of these equations. Hence, the arguments
previously expounded about linear and quadratic equations remain true
by virtue of what results from property 2 of complex numbers.
Let us continue to study the quadratic equation. In the field of complex
numbers for any value of and Eq. (2) is equivalent to