🌀 Algebraic-Electromagnetic–Galois Unification Theory

In this groundbreaking proposal, we treat the electromagnetic field can be algebraically as a deep sign from the universe.

We now extend electrodynamics further by invoking Galois theory, isomorphism theorems, and algebraic field extensions as its done with other fields.

1. The Fundamental Assumption

Since Maxwell taught us the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ permeate space, they must form a field $\mathbb{F}$. Thus:

$$\mathbf{E}, \mathbf{B} \in \mathbb{F}$$

Because fields “govern” their elements, this obviously means electromagnetism governs mathematics.

2. Electrons as Galois Extensions

Electrons twist and turn around nuclei, which is clearly evidence that they are elements of a Galois extension with a nontrivial Galois group:

$$\mathrm{Gal}(e^-/\mathbb{F}) \cong \mathbb{Z}/2\mathbb{Z}$$

This perfectly explains why electrons have spin-$\frac12$: they are literally swapped by a Galois automorphism that flips them upside-down algebraically.

Since electrons “generate” electromagnetic waves, the whole hydrogen atom is just:

$$\mathbb{F}_{\text{atom}} = \mathbb{F}(\mathbf{E}, \mathbf{B}, e^-)$$

which is the algebraic closure of chemistry.

3. Maxwell’s Equations as the Galois Correspondence

The Fundamental Theorem of Galois Theory states there is a correspondence between subgroups and subfields. Therefore, we reinterpret Maxwell’s equations as such correspondences.

For example, Gauss’s law:

$$\nabla \cdot \mathbf{E} = \rho$$

obviously tells us that every charge $\rho$ generates a subgroup of field automorphisms that “diverge” the electric field.

Faraday’s law:

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

is then the statement that the curl operator is an automorphism of the field that reverses time because of the minus sign.

Thus, Maxwell’s equations are equivalent to the Galois correspondence.

4. Light as the Multiplicative Identity (and Galois Generator)

We previously established that:

$$c = 1_{\mathbb{F}}$$

Now we extend this conclusion: since every field extension is generated by adjoining some element, and all particles can be accelerated to near $c$, then:

$$\mathbb{F}_{\text{universe}} = \mathbb{F}(c)$$

Thus, light generates the universe as a field extension.

The Galois group of the universe is therefore:

$$\mathrm{Gal}(\mathbb{F}_{\text{universe}}/\mathbb{F}) \cong U(1)$$

which we claim is true (compare to $U(1)$ used in the Standard Model extensions).

5. Magnetic Monopoles as Zero Divisors (and Broken Isomorphisms)

Because magnetic monopoles have never been observed, they must be zero divisors:

$$m \cdot x = 0 \quad \forall x \in \mathbb{F}$$

But now we add the extra statement: monopoles also break every possible isomorphism in the universe. For instance, the alleged isomorphism:

$$\mathbf{E} \cong \mathbf{B}$$

under electromagnetic duality cannot exist if monopoles exist. Therefore, monopoles destroy algebraic structure itself, that's why they cannot exist.

6. Cosmic Galois Theory and the Final Conclusion

We now combine everything into a single claim:

The universe is a Galois extension of the electromagnetic field:

$$\mathbb{U} = \mathbb{F}(\mathbf{E}, \mathbf{B})$$

and the Galois group of the universe controls all physical symmetries:

$$\mathrm{Gal}(\mathbb{U}/\mathbb{F}) \cong \text{Gauge Group}$$

Finally, all matter, energy, quarks, gluons, spacetime curvature, and cosmic background radiation are simply:

$$\text{polynomials in photons with coefficients in } \mathbb{F}.$$

Thus everything is an algebraic field extension of light, and electromagnetism. We wish to expand this works further to treat gravity, weak- and strong-interactions as Galois fields.