In this first section we recall some basic concepts about complex numbers. A deep knowledge of these notions is essential for the comprehension of the following topics of the course.
The need of extending the real field has its origins in the impossibility of finding a solution of particular equations, such as:
for which it is clear that .
We define the complex field , so that we can solve equations such as the previous one.
is defined through the extension of the well-known properties of .
We remember that is a vector field if it is equipped with the following operations:
defined in an axyomatic way, in order to satisfy the following properties:
- Commutative property: we have that:
- Associative property: we have that:
- Distributive property: we have that:
- Identity element: we have that:
- Inverse element (addition):
- Inverse element (multiplication):
We define the complex field as the set of the pairs with equipped with the properties:
we see that if we define we have that:
- is a field, with theoperations and properties inherited by
- is isomorphic to that is:
Moreover, it can be proved that the function:
is an isomorphism, that is .