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Galois Theory of Difference Equations by Marius van der Put download in pdf, ePub, iPad

After the discovery

After the discovery of Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his solution in his Ars Magna. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degree. Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree. Originally, the theory has been developed for algebraic equations whose coefficients are rational numbers. Crucially, however, he did not consider composition of permutations.

This group was always solvable

In Germany, Kronecker's writings focused more on Abel's result. Nature of the roots for details. It was Rafael Bombelli who managed to understand how to work with complex numbers in order to solve all forms of cubic equation.

With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case, but Cardano did not know this.