1.1 Fields
This book assumes that the reader is familiar with elementary algebra of real and complex numbers.
Properties of Algebra
Let denote either the set of real numbers or the set of complex numbers.
1. Closure under Addition: For any two elements :
2. Closure under Multiplication: For any two elements :
3. Commutativity of Addition: For any two elements :
4. Associativity of Addition: For any three elements :
5. Additive Identity: There exists an element such that for any :
6. Additive Inverse: For each element , there exists an element such that:
7. Commutativity of Multiplication: For any two elements :
8. Associativity of Multiplication: For any three elements :
9. Multiplicative Identity: There exists an element such that for any :
10. Multiplicative Inverse: For each non-zero element , there exists an element such that:
Field
Suppose there is a set of elements , , , … with two operations, addition and multiplication, that satisfy the properties 1-10 of algebra mentioned above. Then is called a field.
With the usual operations of addition and multiplication, the set of complex numbers is a field. The set of real numbers is also a field.
A subfield of the field is a set of complex numbers which is itself a field under the usual operations of addition and multiplication of complex numbers.