The core of this unusual book is the first complete English translation of Galois’ ’Memoir on the conditions for solvability of equations by radicals’, written in 1831 and revised on the eve of the fatal duel in 1832. The author first establishes the background of Galois’ work, including the resolvents, of Vandermonde and Lagrange and Gauss’ solution of the cyclotomic equation. He then explains Galois’ paper in terms understandable to mathematicians of today, completing the proofs which Galois left to the reader. He also fills in the gaps in Galois’ arguments on the existence of a splitting field of a polynomial.
The author takes a strictly constructive stance on the reducibility in an algebraic or transcendental extension of a polynomial irreducible over a ground field. For this he revives and clearly explains the ideas of Kronecker. Thus the approach contrasts strongly with the Dedekind-Artin version of Galois theory that we find in most modern texts. At the same time, the author links the classical and modern approaches; for example he explains the connection between permutations of the roots of a polynomial and automorphisms of the splitting field; and between the composition of substitutions and abstract group theory. Some of this occurs in the admirable collection of exercises, all of which are completely solved. This may lessen the value of the book as a text on field theory, but certainly enhances it as a reference that belongs in the library of every mathematician with some taste for the history of his or her subject.