POLYNOMIALS BELONGING TO GALOIS GROUP AND IT’S SOLVABILITY PROBLEMS

Abstract

The fundamental focus of field theory is the investigation of algebraic extensions of fields, while the investigation of the structures of algebra known as groups is what is covered in group theory. The Galois theory is a branch of algebraic mathematics that studies the connection between these two other subfields of the subject. Everest Galois, a French mathematician, is credited with being the first person to have the idea for it, which he had around the turn of the 19th century. Galois theory offers a complete view of the solutions that may be found for polynomial equations and investigates the symmetries that are already present in these solutions. This view can be used to solve polynomial problems. This mathematical concept was given its name in honor of the French mathematician Roger Galois. A connection between field extensions and the subgroups of the Galois group that is associated with the extension is forged as a direct result of the aforementioned fact, which links the extension. The Galois group helps in the categorization of the solutions to polynomial equations and is responsible for capturing the symmetries of the field extension. Additionally, it helps with the organization of the solutions into categories.