The Unsolvability of a Fifth Degree Polynomial and Related Galois Theory
McCleery, John Arthur
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In retrospect, this paper has been divided into two related parts. First, after a preliminary example, we considered the fundamental theorem of Galois theory and related concepts, among them: a) extensions, b) normal extensions c) splitting fields d) fixed fields e) automorphism groups We then continued in the second and perhaps main task of the paper to show that there is a fifth degree polynomial which is not solvable by radicals. We end by reconsidering the method we used to obtain this result: a) We defined solvability by radicals. b) We defined solvable groups. c) We showed that S5 is not solvable. d) We showed that if p(x) is solvable by radicals, then the automorphism group of p(x) is a solvable group. e) We showed that there is a fifth degree polynomial whose automorphism group is S5.