University Mathematics Handbook (2015)

X. Algebra

Chapter 2. Fields

a.  A field is a set of at least two terms, in which two closed operations, addition and multiplication, are defined. That is, for every and of the field, and , are of the field, and have the following properties:

1.  Addition is commutative: .

2.  Addition is associative: .

3.  There exists one element, , indifferent to addition. That is, for very element of .

4.  For each there is one element, , called additive inversed, such that .

5.  Multiplication is commutative: .

6.  Multiplication is associative:

7.  There exists one element, , indifferent to multiplication. That is, for every of .

8.  For every there is one element , such that .

9.  Multiplication is distributive with respect to addition: .

The terms of a field are called scalars.

b.   is the real numbers field.

c.   is the complex numbers field.

d.   is the rational numbers field.