Definition – Field.

Definition

Let $A$ be any set of any type of object, and let $+$ and $\times$ be two operations which we may define. The set $A$ together with the two operations $+, \times$ are called a field if they satisfy the following field axioms,

To every , there corresponds one element , where , called the sum of such that,
1. The $+$ operator is commutative: $x+y=y+x$ .
2. The operator $+$ is associative: $x+(y+s) = (x+y)+s$ .
3. There exists a unique element $0\in{A}$ , called zero (which again we may define), such that: $x+0 = x$ .
4. For every $x\in{A}$ there is a correspdonding element $-x\in{A}$ such that: $x+(-x) = 0$ .
To every , there corresponds one element , where , called the product of such that,
1. The operator $\times$ is commutative: $x\times{y} = y\times{x}$ .
2. The operator $\times$ is associative: $x\times{(y\times{s})} = (x\times{y})\times{s}$ .
3. There exists a unique element $1\in{A}$ where $1\neq 0$ , such that: $1\times x = x$ .
4. For every element $x$ , $x\neq{0}$ , there is a corresponding element $x^{-1}\in{A}$ such that $x\times{x^{-1}} = 1$ .
The operator $\times$ is distributive with respect to the $+$ operator: $x\times{(y+s)}=x\times{y}+x\times{s}$ .

Notes