P^2+Σ2P

Twin prime seems to be related to Galois theory. Galois described field extension and intermediate field. F=coefficient, K=field extension, M=intermediate field. G=Galois group. Therefore F⊂M⊂K G(K/M) G(K/F) ∴ G(K/M)⊂G(K/F) There is cubic equation. x^3+ax+b=(x-α)(x-β)(x-γ)=0 α+β+γ=0, αβ+βγ+αγ=a, αβγ=-b G e f1 f2 g1 g2 g3 α α β γ β α γ β β γ α α γ β γ γ α β γ β α e⊂H⊂G H1=(e,g1) H2=(e,g2) H3=(e,g3) H=(e,f1,f2) H e f1 f2 α α β γ β β γ α γ γ α β I define F⊂M⊂K and G(K/M)⊂G(K/F) Therefore, G=F and H=M. K⊂M⊂F This is upside-down. ∴ K(G(M))=M, G(K(H))=H

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.