Fundamentals of Galois theory.

Abstract

Galois Theory gives a necessary and sufficient condition for the solvability of polynomials by the elementary arithmetic operations and extraction of roots. The problems of trisecting angles and constructing polynomials with a specific number of sides are also answered in the field of Galois Theory. The purpose of this thesis is to establish the Fundamental Theorem of Galois Theory. This theorem shows the existence of a one-to-one correspondence, that reverses inclusion, between the subfields of a finite normal separable extension of a base field of characteristic zero and the subgroups of the automorphism group of the extension field which fix the base field. We also examine the concepts of algebraic extensions and splitting fields of irreducible polynomials.