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

Chapter 2
The complex numbers
When studying mathematics in school, the set of numbers considered was
progressively extended. The reason for this was based on these extensions
allowing us to operate on numbers with more freedom. So on passing from
the natural numbers to the integers it became possible to subtract any two
numbers; on passing to the rational numbers it became possible to divide
any two numbers, etc.. But the most important result of such extensions
consists in the properties of the extended system often allowing us to
discover some new properties of the initial system. For example, many
difficult problems of number theory, concerning only integers, were solved
using the real numbers as well as the complex numbers.
Historically, the complex numbers appeared just as a way of solving
certain problems in the real numbers. So, for example, the Italian
mathematician Cardano (1501–1576) devised a correct procedure for determining
the roots of the equation of third degree using, in intermediate
steps of calculations, the ‘non-existing’ roots of negative numbers.
Afterwards the complex numbers played an increasingly important
role in mathematics and applications. They were introduced for the first
time in the theory of algebraic equations, because the domain of complex
numbers turned out a more convenient setting for the study of such
equations.
For example, every algebraic equation of degree with real or
complex coefficients has at least one complex root (see below the ‘fundamental
theorem of algebra’ §2.8) whereas not all algebraic equations with
real coefficients have at least one real root.
Since an interpretation of complex numbers was found in terms of
vectors in the plane, geometrical notions such as that of continuity and
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