Intermediate Field

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

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.