Overview

We introduce complex numbers; that is numbers of the form a+ bi where the number satisfies ²= ?1. Mathematicians were led to complex numbers in their efforts to solve so-called algebraic equations; that is equations of the form a _n x ⁿ+ a _{n -1} x ^{n -1}+...+ a ₀=0, with a _k ? , n ? . Our introduction follows the same idea although in a modern mathematical setting. Complex numbers now play important roles in physics, hydrodynamics, electromagnetic theory, electrical engineering as well as pure mathematics.

7.1 Field Extensions

During the history of civilisation the concept of a number was unceasingly extended, from integers to rationals, from positive numbers to negative numbers, from rationals to reals, etc. We now embark on an extension of reals to a field in which the equation

(7.1)

has a solution. This will be the field of complex numbers.

Let us consider the following question. Is it possible to extend the field of rationals to a larger field in which the equation ? ² ?2=0 is solvable? The obvious answer is yes: the reals. Is there a smaller field? The answer is again yes: there is a smallest field which contains rationals and , namely the intersection of all fields which contain and the (real) number . Can this field be constructed directly without using the existence of ? The answer is contained in Exercises 7.1.1-7.1.3.

The process of extension of a...