Field Theory and Its Classical Problems lets Galois theory unfold in a natural way, beginning with the geometric construction problems of antiquity, continuing through the constructability of regular n-gons and the properties of roots of unity, and then on to the solvability of polynomial equations by radicals, and beyond.
The Three Greek Problems
Solution by Radicals
Polynomials with Symmetric Groups
Solutions to the Problems
Charles Hadlock received his Ph.D. in mathematics from the University of Illinois in 1970. He has studied collapse processes from many points of view: from one career as a world traveling consultant with Arthur D. Little, Inc., working to head off catastrophic risks in the chemical, power, transportation, and mining industries; to another as the Dean of a business school witnessing collapses of corporations, currencies, and markets. Add to this his broad mathematical and scientific background; collaborative work with political scientists, engineers, and others; and extensive experience with management challenges at the top levels of corporations and governments. He is an award winning author with a Carus Monograph on field theory and an acclaimed text on mathematical modeling in environmental management. He has served on the mathematics faculties of Amherst and Bowdoin Colleges, as a Visiting Professor of Earth, Atmospheric, and Planetary Sciences at MIT, and is currently Trustee Professor of Technology, Policy, and Decision Making at Bentley University.
Hadlock's book sports one of the best prefaces I've ever read in a mathematics book. The rest of the book is even better: in 1984 it won the first MAA Edwin Beckenbach Book Prize for excellence in mathematical exposition.
Hadlock says in the preface that he wrote the book for himself, as a personal path through Galois theory as motivated by the three classical Greek geometric construction problems (doubling the cube, trisecting angles, and squaring the circle — all with just ruler and compass) and the classical problem of solving equations by radicals. Unlike what happens in most books on the subject, all three Greek problems are solved in the first chapter, with just the definition of field as a subfield of the real numbers, but without even defining degree of field extensions, much less proving its multiplicativity (this is done in chapter 2). Continued...