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Field Theory and Its Classical Problems
 
Field Theory and Its Classical Problems
MAA Press: An Imprint of the American Mathematical Society
This title is not currently available
Field Theory and Its Classical Problems
Click above image for expanded view
Field Theory and Its Classical Problems
MAA Press: An Imprint of the American Mathematical Society
This title is not currently available
  • Book Details
     
     
    The Carus Mathematical Monographs
    Volume: 191975; 323 pp
    MSC: Primary 12

    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.

  • Table of Contents
     
     
    • Front Cover
    • Field Theory and its Classical Problems
    • Copyright Page
    • Preface
    • Contents
    • Introduction
    • Chapter 1—The Three Greek Problems
    • 1.1. Constructible Lengths
    • 1.2. Doubling the Cube
    • 1 3 Trisecting the Angle
    • 1.4. Squaring the Circle
    • 1.5. Polynomials and Their Roots
    • 1.6. Symmetric Functions
    • 1.7. The Transcendence of π
    • References and Notes
    • Chapter 2—Field Extensions
    • 2.1. Arithmetic of Polynomials
    • 2.2. Simple, Multiple, and Finite Extensions
    • 2.3. Geometric Constructions Revisited
    • 2.4. Roots of Complex Numbers
    • 2.5. Constructibility of Regular Polygons I
    • 2.6. Congruences
    • 2.7. Constructibility of Regular Polygons II
    • References and Notes
    • Chapter 3—Solution by Radicals
    • 3.1. Statement of the Problem
    • 3.2. Automorphisms and Groups
    • 3.3. The Group of an Extension
    • 3.4. Two Fundamental Theorems
    • 3.5. Galois' Theorem
    • 3.6. Abel's Theorem
    • 3.7. Some Solvable Equations
    • References and Notes
    • Chapter 4—Polynomials with Symmetric Groups
    • 4.1. Background Information
    • 4.2. Hubert's Irreducibility Theorem
    • 4.3. Existence of Polynomials over Q with Group Sn
    • References and Notes
    • Solutions to the Problems
    • Index
  • Additional Material
     
     
  • Reviews
     
     
    • 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.
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 191975; 323 pp
MSC: Primary 12

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.

  • Front Cover
  • Field Theory and its Classical Problems
  • Copyright Page
  • Preface
  • Contents
  • Introduction
  • Chapter 1—The Three Greek Problems
  • 1.1. Constructible Lengths
  • 1.2. Doubling the Cube
  • 1 3 Trisecting the Angle
  • 1.4. Squaring the Circle
  • 1.5. Polynomials and Their Roots
  • 1.6. Symmetric Functions
  • 1.7. The Transcendence of π
  • References and Notes
  • Chapter 2—Field Extensions
  • 2.1. Arithmetic of Polynomials
  • 2.2. Simple, Multiple, and Finite Extensions
  • 2.3. Geometric Constructions Revisited
  • 2.4. Roots of Complex Numbers
  • 2.5. Constructibility of Regular Polygons I
  • 2.6. Congruences
  • 2.7. Constructibility of Regular Polygons II
  • References and Notes
  • Chapter 3—Solution by Radicals
  • 3.1. Statement of the Problem
  • 3.2. Automorphisms and Groups
  • 3.3. The Group of an Extension
  • 3.4. Two Fundamental Theorems
  • 3.5. Galois' Theorem
  • 3.6. Abel's Theorem
  • 3.7. Some Solvable Equations
  • References and Notes
  • Chapter 4—Polynomials with Symmetric Groups
  • 4.1. Background Information
  • 4.2. Hubert's Irreducibility Theorem
  • 4.3. Existence of Polynomials over Q with Group Sn
  • References and Notes
  • Solutions to the Problems
  • Index
  • 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.
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.