The construction of solvable polynomials

Abstract:

Although Leopold Kronecker’s 1853 paper “On equations that are algebraically solvable” is famous for containing his enunciation of the Kronecker-Weber theorem, its main theorem is an altogether different one, a theorem that reduces the problem of constructing solvable polynomials of prime degree $\mu$ to the problem of constructing cyclic polynomials of degree $\mu -1$. Kronecker’s statement of the theorem is sketchy, and he gives no proof at all. There seem to have been very few later treatments of the theorem, none of them very clear and none more recent than 1924. A corrected version and a full proof of the theorem are given. The main technique is a constructive version of Galois theory close to Galois’s own.