Field/Seven elements/Introduction/Example

On the set ${\displaystyle {}\{0,1,2,3,4,5,6\}}$ (with seven elements), one can define a field structure using

${\displaystyle {}+}$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$
${\displaystyle {}0}$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$
${\displaystyle {}1}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$	${\displaystyle {}0}$
${\displaystyle {}2}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$	${\displaystyle {}0}$	${\displaystyle {}1}$
${\displaystyle {}3}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$
${\displaystyle {}4}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$
${\displaystyle {}5}$	${\displaystyle {}5}$	${\displaystyle {}6}$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$
${\displaystyle {}6}$	${\displaystyle {}6}$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$

${\displaystyle {}\cdot }$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$
${\displaystyle {}0}$	${\displaystyle {}0}$	${\displaystyle {}0}$	${\displaystyle {}0}$	${\displaystyle {}0}$	${\displaystyle {}0}$	${\displaystyle {}0}$	${\displaystyle {}0}$
${\displaystyle {}1}$	${\displaystyle {}0}$	${\displaystyle {}1}$	${\displaystyle {}2}$	${\displaystyle {}3}$	${\displaystyle {}4}$	${\displaystyle {}5}$	${\displaystyle {}6}$
${\displaystyle {}2}$	${\displaystyle {}0}$	${\displaystyle {}2}$	${\displaystyle {}4}$	${\displaystyle {}6}$	${\displaystyle {}1}$	${\displaystyle {}3}$	${\displaystyle {}5}$
${\displaystyle {}3}$	${\displaystyle {}0}$	${\displaystyle {}3}$	${\displaystyle {}6}$	${\displaystyle {}2}$	${\displaystyle {}5}$	${\displaystyle {}1}$	${\displaystyle {}4}$
${\displaystyle {}4}$	${\displaystyle {}0}$	${\displaystyle {}4}$	${\displaystyle {}1}$	${\displaystyle {}5}$	${\displaystyle {}2}$	${\displaystyle {}6}$	${\displaystyle {}3}$
${\displaystyle {}5}$	${\displaystyle {}0}$	${\displaystyle {}5}$	${\displaystyle {}3}$	${\displaystyle {}1}$	${\displaystyle {}6}$	${\displaystyle {}4}$	${\displaystyle {}2}$
${\displaystyle {}6}$	${\displaystyle {}0}$	${\displaystyle {}6}$	${\displaystyle {}5}$	${\displaystyle {}4}$	${\displaystyle {}3}$	${\displaystyle {}2}$	${\displaystyle {}1}$

Without any further theory, it is very tedious to so show that this is indeed a field.