Field Theory and Galois Theory

Abstract

The general solutions of linear and quadratic polynomial in one variable were known centuries before. For cubic and quartic equations also the general solutions are provided
by Cardano's and Ferrari's methods respectively. In 19th century a great work has been done to find general solution of a general polynomial by radicals. However there was no
success even after efforts of many great mathematicians of that time. Eventually work by Able and Galois gives satisfactory solution and complete understanding of this problem.Galois Theory provides a connection between Field theory and Group theory, which in turn useful to convert problems in field theory into Group theory, which are better
understood and easy to handle. Galois theory not only provide answer to the problem discussed above but also explains why the general solution exists for polynomials with degree less then or equal to 4. In his original work, Galois used permutation groups to describe relations between roots of the polynomial. In modern approach, developed by Artin, Dedekind etc., involves study of
automorphisms of field extensions.