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

The complex numbers 51
Consequently the expression within brackets is the quotient of the division
of the polynomial by the polynomial and is the
remainder. The procedure of the division of two polynomials described
here is called the Euclidean algorithm.
201. Let
for which the degrees of and are lower than the degree of
(may be or Prove that
2.2 The field of complex numbers
From the solution of Problem 194 it follows that there exist fields smaller
than the field of the real numbers; for example, the field of the rational
numbers. We now construct a field which is bigger than the field of the
real numbers: the field of the complex numbers.
Consider all the possible pairs of real numbers, i.e., the pairs of type
where and are two arbitrary real numbers. We will say that
if and only if and In the set of all these pairs
we define two binary operations, the addition and the multiplication, in
the following way:
(here within brackets in the second members of the equations the operations
are the usual operations on real numbers). For example, we obtain
DEFINITION. The set of all pairs of real numbers with the operations
of addition and of multiplication defined by (2.2) and (2.3) is called the
set of complex numbers.