Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$.
If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then we get a finite abelian ring of order $p^2$.
Every integer element $t$ of this ring different from 0 satisfies $t^{p-1} = 1$ ( Fermat's little ).
Likewise every non-integer element $s$ of this ring satisfies $s^{(p-1)^2} = 1$.
The question now becomes if we put the elements in a matrix-like-square with the rules
1) 0 at bottom left
2) +1 means go to right
3) +$b$ means go up
Then is there a pattern in the $k$ th powers of the elements s i.e. $s^k$ ?
With pattern i mean something like Fermat's little or similar algebra or a geometric pattern such as knight moves on a chess board.
I do not believe these patterns to be random.
I have been thinking about the distributive property to get an answer but with no succes sofar.
Sieving also comes to my mind.
How to handle this ?
Also Im not sure about the name abelian ring... I can only find texts about abelian groups most of the time.
Also Im unconfident about notation.
$mod$ $p$ $mod$ $pb$ or $mod$ $pb$ $mod$ $p$ seems weird.
Maybe I just did too many magic knight tours :)