Mod 01(G,∙)(G, \bullet)

Groups

Symmetry

The study of reversibility. Rotations, reflections, and the Rubik's Cube.

Mod 02ϕ(xy)=ϕ(x)ϕ(y)\phi(xy)=\phi(x)\phi(y)

Maps

Relation

Structure-preserving functions. The translation layer between algebraic worlds.

Mod 03(R,+,∙)(R, +, \bullet)

Rings

Arithmetic

Sets with Addition and Multiplication. Where integers and polynomials live.

Mod 04F=R/M\mathbb{F} = R / M

Fields

Continuum

Rings where division works. The foundational scalars of Linear Algebra.

Mod 05V over FV \text{ over } \mathbb{F}

Vector Spaces

Linearity

Abelian groups scaled by a Field. The abstract version of grid-based space.

Mod 06Gal(E/F)\text{Gal}(E/F)

Galois Theory

Unification

The bridge between Groups and Fields. Proves why some equations are unsolvable.