Galois Theory for Beginners
Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. In this book, Bewersdorff follows the historical development of the theory, emphasizing concrete examples along the way. As a result, many mathematical abstractions are now seen as the natural consequence of particular investigations. Few prerequisites are needed beyond general college mathematics, since the necessary ideas and properties of groups and fields are provided as needed. Results in Galois theory are formulated first in a concrete, elementary way, then in the modern form. Each chapter begins with a simple question that gives the reader an idea of the nature and difficulty of what lies ahead. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular $n$-gons are also presented. This book is suitable for undergraduates and beginning graduate students.
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Chapter 1 Cubic Equations
The Birth of the Complex Numbers
Chapter 3 Biquadratic Equations
Chapter 4 Equations of Degree n and Their Properties
Chapter 5 The Search for Additional Solution Formulas
Chapter 6 Equations That Can Be Reduced in Degree
Chapter 7 The Construction of Regular Polygons
adjoined adjunction algorithm Aut(L automorphisms basic arithmetic operations belong biquadratic calculation Cardano Cardano's formula complex numbers computing corresponding cosets cubic equation cyclic cyclotomic equation decomposition defined definition determined divisible elementary symmetric polynomials equa Évariste Galois example expressed extension field field extension fifth degree finite fundamental theorem Galois group Galois resolvent Galois theory Gauss group G group table identity integer coefficients intermediate field intermediate values investigation irreducible known quantities Lagrange resolvent linear factors mathematical multiplication negative numbers nonzero normal subgroup nth root number of elements obtains original equation permutations polynomial f(X previous chapter prime number problem proof properties prove quadratic equation quotient group rational coefficients rational numbers real numbers representation residue classes result roots of unity solution formula solutions a1 solvable in radicals splitting field square roots straightedge and compass Tartaglia theorem of Galois three solutions tion transformation Vandermonde variables Z/nZ zero