On insolvability by radicals of polynomials of degree at least 5 - Leyson, Jeramie C. (College of Science, University of the Philippines Cebu, 2022)
Solvability of polynomials by radicals means that the roots of the polynomial can be expressed in algebraic terms (sums, products, roots) of its coefficients. In this study, we discuss some important concepts of Galois Theory that we use to provide a detailed presentation of a solution to the problem about solvability of polynomials. We construct a detailed proof for the insolvability by radicals of polynomials in an indeterminate x over the field of rational numbers, Q, that are of prime order. This is then followed by a detailed discussion of the general case - the insolvability by radicals of polynomials of degree greater than or equal to 5 over an arbitrary field K of characteristic 0. The proofs presented in this paper are based mainly on the proof provided by John M. Howie. Specific examples are provided to illustrate the procedure suggested by the proof.