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# Fields.

Definition 2.0.1   Let F be a set with two binary operations + and . Suppose that the following properties hold:
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.

Next: Systems of linear equations. Up: No Title Previous: Vectors in Geometry.
Noah Dana-Picard
2001-02-26