Galois theory is one of the most beautiful connections in mathematics, linking the solvability of polynomial equations to the structure of groups. For centuries, mathematicians found formulas for quadratic, cubic, and quartic equations. But the quintic equation remained unsolved until Évariste Galois revolutionized our understanding by showing that some equations simply cannot be solved using radicals.
Field extensions are the foundation of Galois theory. A field is a mathematical structure with addition and multiplication operations. We start with familiar fields like the rationals Q, then create extensions by adjoining new elements. For example, Q(√2) contains all numbers of the form a + b√2 where a and b are rational. Similarly, Q(∛2) contains the cube root of 2. These nested structures show how fields can contain other fields as subfields.