The Algorithmic Foundations of Galois’s Theory

This chapter is devoted to a full exposition of Galois theory, which here means the theory Galois himself explained in his memoir on the conditions for the solvability of equations by radicals, often called his First Memoir. A careful reading of the First Memoir shows that Galois himself thought very algorithmically and sought to approach the problem of solving algebraic equations in a thoroughly constructive way. Early essays in the chapter describe his construction of a splitting field of the polynomial whose roots are to be found, with explicit examples in the cubic case. Later essays show how Galois’s results can be interpreted as an algorithm that constructs a splitting field, using only radical adjunctions, for any polynomial whose Galois group is solvable. The final essay uses this to derive Cardano’s formulas for the roots of a cubic polynomial in terms of radicals.