An Introduction to Galois Theory (with connections to AIT)

This note aims to demystify the foundational definitions of Galois Theory: field extensions, automorphisms, and the Galois group. Rather than stating abstract definitions upfront, we derive them as necessary consequences of algebraic consistency. We begin with the fundamental example of $x^2+1=0$ to illustrate how algebraic symmetries arise from the indistinguishability of roots relative to a base field (e.g., $i$ and $-i$ are merely distinct labels for indistinguishable algebraic twins). Finally, we examine why the Galois group is rarely the full set of permutations, but rather a structure constrained by the ``hidden'' algebraic relations within the field. We connect Galois theory to the perspective of mathematical structure, symmetry, and AIT.