Galois Theory is a branch of mathematics that connects field theory and group theory, providing a profound understanding of the solvability of polynomial equations by analyzing the symmetries of their roots through groups. It investigates how the structure of a field extension relates to the structure of the Galois group, which consists of field automorphisms that fix the base field. This theory has far-reaching implications in various areas of mathematics, including the study of solvable and nilpotent groups, as well as the applications of these concepts in solving polynomial equations.
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