Homomorphic

I describe symmetry which is based on Galois theory.
I think that it must be related to prime numbers.

Quadratic equation is like a mirror. There is the formula.











f(x)=0 have two solutions, α and β.


∴α+β=-a

As you know, f(α)=f(β)=0.

If f(α)=β and f(β)=α, f(f(α))=α and f(β)=α.

I define α+β=-a.

Therefore

f(α+β)=f(-a)
f(α)+f(β)=β+α=-a
β+f(β)=-a
∴f(β)=-a-β=α+β-β=α.

There is symmetry.