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

46 Chapter 2
geometrical transform became applicable to the study of complex numbers.
The relation between complex numbers and vectors also allows us to
rewrite several problems of mechanics in terms of complex numbers and
their equations — in particular, in hydrodynamics and aerodynamics, the
theory of electricity, thermodynamics, etc..
2.1 Fields and polynomials
Real numbers can be added, multiplied, and the inverse operations are
also allowed: the subtraction and the division (the latter, however, not
by zero). In any addition of several numbers the terms can be permuted
in any way, and they can be collected arbitrarily within brackets without
changing the result. The same holds for the factors of any product. All
these properties, as well as the relation between the addition and the
multiplication, can be summarized as it follows:
The real numbers possess the three following properties:
1) They form a commutative group (see §1.3) under addition (the unit
element of this group is denoted by 0 and is called the zero).
2) If one excludes 0 then the real numbers form a commutative group
under multiplication.
3) The addition and the multiplication are related by distributivity:
for any numbers
The existence of these three properties is very important because they
allow us to simplify the arithmetic of algebraic expressions, to solve equations,
etc.. The set of real numbers is not the only set to possess these
three properties. In order to single out all these sets the following notion
is introduced.
DEFINITION. A set in which two binary operations (addition and
multiplication) possessing the above properties are defined is called a
field.
194. Verify whether the following subsets of the real numbers set
with the usual operations of addition and multiplication are a field: a)
all the natural numbers; b) all the integer numbers; c) all the rational
numbers; d) all the numbers of the type where and are
two arbitrary rational numbers.