- 1.
- Addition:
- S 1.
- Closure:
and
*a*+*b*is uniquely determined. - S 2.
- Associativity: .
- S 3.
- Identity (neutral element): There exists an element
such that 0 +
*a*=*a*and*a*+ 0=*a*for all . This element 0 is called the*identity*of*F*for the addition. - S 4.
- Opposites: For each ,
there exists an element
such that
*a*+*b*=*b*+*a*=0. The lement*b*is called the*ooposite*element of*a*and is denoted by -*a*.

- 2.
- Multiplication:
- M 1.
- Closure: and is uniquely determined.
- M 2.
- Associativity: .
- M 3.
- Identity (neutral element): There exists an element
such that
and
for all .
This element 1 is called the
*identity*of*G*for the multiplication. - M 4.
- Inverses: For each
distinct from 0, there exists an element
such that
.
We denote
*b*=*a*^{-1}; it is called the*inverse element*of*a*in*F*.

- 3.
- Distributivity:

The classical examples that we need are , and . We will work most often with real numbers (i.e. over the field , but sometimes we will give examples over too. These last examples are of special interest because of their applications when solving differential equations.