A Brief Summary of Polynomials and Galois Theory

Overview & Synopsis

In these notes, Galois theory is presented as the study of the relation between polynomials, fields, and groups. The three topics are organized around the central problem on the factorization of polynomials after extension of coefficients. The proof of the fundamental theorem of Galois theory is entirely based on dimension counting and completely independent of the notion of polynomials. In terms of polynomials, the fundamental theorem is interpreted as an equivalence between the factorization of separable polynomials and the orbit decomposition of their roots under the action of the Galois group. In the application to solvability by radicals, some of the group theoretic arguments have been demonstrated using an alternative definition of finite solvable groups in terms of group characters.