Vector Space

DEFINITION1: Let $K$ be a set with at least two distinct elements, $+:K\times{K}\rightarrow{K}$ and $\cdot:K\times{K}\rightarrow{K}$ be two functions. If the following conditions hold, then $\left( K,+,\cdot \right)$ is called a field:

F1) $\left( a+b \right) +c=a+ \left( b+c \right)$ for all $a,b,c\in{K}$,

F2) $a+b=b+a$ for all $a,b\in{K}$,

F3) there exists ${0}\in{K}$ such that $a+0=a$ for all $a\in{K}$,

F4) for each $a\in{K}$, there exists $b\in{K}$ such that $a+b=0$,