Galois Theory, third edition
This is the third edition of the classic textbook of Galois theory, first published in 1972. But it is not just a reprint of earlier editions. Those who know the first two editions will be surprised by a radical change of presentation. The author reversed the original Bourbakiste approach expressed by a slogan “from general to concrete” and now presents the theory in the direction “from concrete to general”. Thus after a historical chapter, he starts with solutions in radicals of polynomial equations of degree 2, 3, 4, and presents a quintic equation solvable in radicals. Factorization of complex polynomials is developed from theory of polynomials with complex coefficients and the fundamental theorem of algebra. Field extensions of rational numbers follow including the definition of rational expressions and the degree of an extension. As a digression, the author proves non-existence of ruler-and-compass solutions of classical geometric problems of squaring the cube, trisecting the angle and squaring the circle. Then the Galois theory starts. After a short explanation of Galois groups according to Galois, he presents modern definitions of the Galois correspondence, splitting fields, normal and separable extensions, and field automorphisms. The fundamental Galois correspondence between the subfields of a field extension and subgroups of the automorphism group of the extension is proved. An example of the correspondence resulting from a quartic equation is also given. Solvable and simple groups are introduced and the Galois theorem about solvability of equations in radicals is proved. After all this, abstract rings and fields are introduced and the abstract theory of field extensions is developed. The last part of the book contains some applications, e.g., the construction of finite fields, constructions of regular polygons, circle divisions (including cyclotomic polynomials) and an algorithm on how to calculate the Galois group of a polynomial equation. The penultimate chapter is about algebraically closed fields and the last chapter, on transcendental numbers, contains “what-every-mathematician-should-see-at-least-once”, the proof of transcendence of p. The book is designed for the second and third year undergraduate courses. I will certainly use it.