What is the square root of two? | The Fundamental Theorem of Galois Theory

Automorphisms of field extensions are structure-preserving bijections that fix the base field. These maps formalize the notion of “permutations that preserve algebraic relations” between conjugate elements.

Galois theory studies conjugate numbers—algebraically indistinguishable elements satisfying the same polynomial equations with rational coefficients. These numbers represent different roots of the same minimal polynomial over the rationals.

Galois theorists study field extensions K/F, where a larger field K contains a smaller field F. These extensions provide the setting for understanding algebraic relations between conjugate elements and their symmetries.

The fundamental theorem establishes a bijective correspondence between subgroups of the Galois group and intermediate field extensions. This profound result connects abstract group theory to concrete field extensions, unifying algebra.

The Galois group Gal(K/F) consists of all field automorphisms of K fixing F. This group captures the complete algebraic symmetry of the extension, encoding which permutations of roots preserve polynomial relations.

The Abel-Ruffini theorem, proved using Galois theory, states that general quintic (degree 5) polynomials cannot be solved using radicals (nth roots and arithmetic operations). This resolved a centuries-old quest for a quintic formula analogous to the quadratic formula.