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Fields.

Definition 2.0.1   Let F be a set with two binary operations + and $\cdot$. Suppose that the following properties hold:
1.
Addition:
S 1.
Closure: $\forall a,b \in F, \; a + b \in F$ and a + b is uniquely determined.
S 2.
Associativity: $\forall a,b,c \in F,\; a + (b + c) = (a + b) + c$.
S 3.
Identity (neutral element): There exists an element $0 \in F$ such that 0 + a=a and a + 0=a for all $a \in F$. This element 0 is called the identity of F for the addition.
S 4.
Opposites: For each $a \in F$, there exists an element $b \in F$ 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: $\forall a,b \in F, \; a \cdot b \in F$ and $a \cdot b$ is uniquely determined.
M 2.
Associativity: $\forall a,b,c \in F,\; a \cdot (b \cdot c) = (a \cdot b) \cdot c$.
M 3.
Identity (neutral element): There exists an element $1 \in F$ such that $1 \cdot a=a$ and $a \cdot 1=a$ for all $a \in G$. This element 1 is called the identity of G for the multiplication.
M 4.
Inverses: For each $a \in F$ distinct from 0, there exists an element $b \in F$ such that $a \cdot b = b \cdot a =1$. We denote b=a-1; it is called the inverse element of a in F.
3.
Distributivity:

\begin{displaymath}\forall a,b \in F, \; a \cdot (b + c) = a \cdot b + a \cdot c.
\end{displaymath}

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


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