An Introduction to Galois Theory (with connections to AIT)

    This note aims to demystify Galois Theory by connecting its foundational definitions to a broader principle of \textit{computational and compositional tractability}. We begin with the elementary example of $\mathbb{Q}(i)$ to illustrate how algebraic symmetries arise from the indistinguishability of roots. We then proceed to the heart of the theory: the rigorous link between symmetry and solvability.  We reframe the Abel-Ruffini theorem not merely as a limit on polynomials, but as a positive definition of structure: a problem is "solvable" if and only if its symmetry group is \textit{compositional} (decomposable into abelian steps). When this condition fails—as with the monolithic $A_5$ group—we show that progress requires the discovery of new atomic primitives (the ``Nuclear Option"). Finally, we discuss this logic in the context of the continuous domain (Lie Theory) and Artificial Intelligence. Drawing on Poggio’s work on compositionality and Kolmogorov Theory, we demonstrate that ``learning" is the algorithmic cycle of discovering these symmetries. The resulting ``staircase" in the Kolmogorov Structure Function reveals that intelligence is the interplay of two modes: \textit{exploiting} existing tools through composition, and \textit{inventing} new cognitive cores to further compositionally compress the ``unsolvable".