# Galois' Theory of Algebraic Equations

## Jean-Pierre Tignol

- 324 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# Galois' Theory of Algebraic Equations

## Jean-Pierre Tignol

## About This Book

The book gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel, and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as "group" and "field".

A brief discussion of the fundamental theorems of modern Galois theory and complete proofs of the quoted results are provided, and the material is organized in such a way that the more technical details can be skipped by readers who are interested primarily in a broad survey of the theory.

In this second edition, the exposition has been improved throughout and the chapter on Galois has been entirely rewritten to better reflect Galois' highly innovative contributions. The text now follows more closely Galois' memoir, resorting as sparsely as possible to anachronistic modern notions such as field extensions. The emerging picture is a surprisingly elementary approach to the solvability of equations by radicals, and yet is unexpectedly close to some of the most recent methods of Galois theory.

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The book gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel, and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as "group" and "field".

A brief discussion of the fundamental theorems of modern Galois theory and complete proofs of the quoted results are provided, and the material is organized in such a way that the more technical details can be skipped by readers who are interested primarily in a broad survey of the theory.

In this second edition, the exposition has been improved throughout and the chapter on Galois has been entirely rewritten to better reflect Galois' highly innovative contributions. The text now follows more closely Galois' memoir, resorting as sparsely as possible to anachronistic modern notions such as field extensions. The emerging picture is a surprisingly elementary approach to the solvability of equations by radicals, and yet is unexpectedly close to some of the most recent methods of Galois theory.

Request Inspection Copy

Readership: Upper level undergraduates, graduate students and mathematicians in algebra.

Key Features:

- Describes the problems and methods at the origin of modern abstract algebra
- Provides an elementary thorough discussion of the insolvability of general equations of degree at least five and of ruler-and-compass constructions
- Original exposition relying on early sources to set classical Galois theory into its historical perspective

## Frequently asked questions

## Information

**Chapter 1**

**Quadratic Equations**

*ax*=

*b*does not use anything more than a division, it hardly belongs to the algebraic theory of equations; it is therefore appropriate to begin our discussion with quadratic equations

*ax*

^{2}+

*bx*+

*c*= 0(

*a*≠ 0).

*a*, we reduce to the case where the coefficient of

*x*

^{2}is 1:

*x*

^{2}+

*px*+

*q*= 0.

*x*easily follow:

^{1}Nevertheless, mathematicians had been solving quadratic equations for about 40 centuries before. The purpose of this first chapter is to give a brief outline of this “prehistory” of the theory of quadratic equations.

**1.1Babylonian algebra**

*x*) when the difference between the area and the side (i.e.,

*x*

^{2}−

*x*) is given; in other words, it gives the solution of

*x*

^{2}−

*x*=

*b*.

^{−1}, i.e., 0.5. The square of this 30 is then 15 which means 0.25, and this explains why the sum of 14.30 and 15 is written as 14.30.15: in modern notation, the operation is 870 + 0.25 = 870.25.

*x*

^{2}−

*x*= 870, and gets

*x*= 30. The other solution

*x*= −29 is neglected, since the Babylonians had no negative numbers.

*x*

^{2}+

*ax*=

*b,x*

^{2}−

*ax*=

*b*,and

*x*

^{2}+

*b*=

*ax*,

*a, b*stand for positive numbers. (The fourth type

*x*

^{2}+

*ax*+

*b*= 0 obviously has no (positive) solution.)

*x*and

*y*respectively denote the length and the breadth of the rectangle, this problem amounts to solving the system

*y*, this system yields the following equation for

*x*:

*x*is eliminated instead of

*y*, we get

*y*=

*x*−

*a*or

*x*=

*y*+

*a*.

*z*be the arithmetic mean of

*x*and

*y*.

*z*is the side of the square which has the same perimeter as the given rectangle:

*z*

^{2}) to the area of the rectangle (

*xy*=

*b*). We have

*x*

^{2}−

*ax*=

*b*and

*y*

^{2}+

*ay*=

*b*.

*x*

^{2}+

*b*=

*ax*does not explicitly appear. This is even more puzzling in view of the frequent occurrence in Babylonian tablets of problems such as to find the length and the breadth of a rectangle when the perimeter and the area of the rectangle are given, which amounts to the solution of

*y*, this system leads to

*x*

^{2}+

*b*=

*ax*. So, why did Babylonians solve...