# Field/Two elements/Example

We are trying to find a structure of a field on the set ${\displaystyle {}\{0,1\}}$. If ${\displaystyle {}0}$ is supposed to be the neutral element of the addition and ${\displaystyle {}1}$ the neutral element of the multiplication, then everything is already determined: The equation ${\displaystyle {}1+1=0}$ must hold, since ${\displaystyle {}1}$ has an inverse element with respect to the addition, and since ${\displaystyle {}0\cdot 0=0}$ holds due to fact. Hence the operation tables look like

${\displaystyle {}+}$ ${\displaystyle {}0}$ ${\displaystyle {}1}$
${\displaystyle {}0}$ ${\displaystyle {}0}$ ${\displaystyle {}1}$
${\displaystyle {}1}$ ${\displaystyle {}1}$ ${\displaystyle {}0}$

and

${\displaystyle {}\cdot }$ ${\displaystyle {}0}$ ${\displaystyle {}1}$
${\displaystyle {}0}$ ${\displaystyle {}0}$ ${\displaystyle {}0}$
${\displaystyle {}1}$ ${\displaystyle {}0}$ ${\displaystyle {}1}$

With some tedious computations one can check that this is indeed a field.