the unit circle, tangens and the complex plane

[bhelmes]

A peaceful and pleasent evening for all,

I did not understand clearly the connection between the rational points in the unit circle and the connection to the roots of the primes p=1 mod 4

I know the function f(u,v)=u²+v²=(u+vI)(u-vI)
I can make a reference between the corresponding primitiv pyth. tripple (a,b,c) and the complex lattice (u,v).
I know that the distance from the coordinate origin is the euclidean distance or the c of the prim. pyth. trippel c²=a²+b²
I can transformate a pyth. trippel into a point of the unit circle.

And i know that the primes p=1 mod 4 have the order of (p-1)²
in the complex lattice and
reducing them to the subgroup of the unit circle the order (p-1), if i am right

Can i conclude anything about the complex roots of a prime p,
when i transform the pyth. trippel a,b,c with c < p
to the complex lattice with (a/c)²+(b/c)² mod p

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As i guess that the mathematical expression is not clear for everyone,
i add a website with the corresponding values

http://devalco.de/poly_xx+yy_demo.ph...40&radius_c=41

you can choose a prime and you get the (u,v),(a,b,c) and the order in red of the choosen prime

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Is it possible to predict a complex root of p ?

Greetings from the law of tangens
Bernhard