# Field Theory and Its Classical Problems

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*Charles Robert Hadlock*

MAA Press: An Imprint of the American Mathematical Society

Reprinted edition available:
CAR/35

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
construction 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 logical pathway is historic, but the
terminology is consistent with modern treatments. No previous
knowledge of algebra is assumed. Notable topics treated along this
route include the transcendence of \(e\) and \(\pi\),
cyclotomic polynomials, polynomials over the integers, Hilbert's
irreducibility theorem, and many other gems in classical
mathematics. Historical and bibliographical notes complement the text,
and complete solutions are provided to all problems.

#### Reviews & Endorsements

The presented book is a clear and concise introduction to classical results of Galois theory. The book is an excellent reading for everyone, especially for instructors and first year graduate students in Galois theory.

-- Acta. Sci. Math.

# Table of Contents

## Field Theory and Its Classical Problems

- Front Cover Cover11
- Field Theory and its Classical Problems v6
- Copyright Page vi7
- Preface ix10
- Contents xv16
- Introduction 118
- Chapter 1—The Three Greek Problems 926
- Chapter 2—Field Extensions 5976
- 2.1. Arithmetic of Polynomials 5976
- 2.2. Simple, Multiple, and Finite Extensions 7188
- 2.3. Geometric Constructions Revisited 8097
- 2.4. Roots of Complex Numbers 84101
- 2.5. Constructibility of Regular Polygons I 92109
- 2.6. Congruences 98115
- 2.7. Constructibility of Regular Polygons II 104121
- References and Notes 119136

- Chapter 3—Solution by Radicals 123140
- Chapter 4—Polynomials with Symmetric Groups 181198
- Solutions to the Problems 221238
- Index 319336