13. Solvability of Equations

In this chapter, we show that equations solvable by radicals are characterized by the solvability of their Galois groups. This immediately implies that general equations of degree 5 and above are not solvable by radicals. If one has a more modest goal to prove that the fifth degree general equation over a number field is not solvable by radicals, then there exists a simple argument by Nagell which only requires limited knowledge of field extensions and no knowledge of Galois theory. We consider Nagell’s proof in the exercises. This chapter further outlines Weber’s theorem on irreducible equations of prime degree (at least 5) with only two nonreal zeros, which are examples of non-solvable equations. We further discuss Galois’ classical theorem, which gives a characterization of irreducible solvable polynomials of prime degree. Both Galois’ and Weber’s results give examples of concrete unsolvable polynomials over the rational numbers. The solvability by real radicals in connection with “casus irreducibilis” is also discussed.