PREFACE :
This is a small book on algebra where the stress is laid on the structure of
fields, hence its title.
You will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. For example, it has been known for centuries how to explicitely solve polynomial equations of degree 2 (Babylonians, many centuries ago), 3 (Scipione del Ferro, Tartaglia, Cardan, around 1500 a.d.), and even 4 (Cardan, Ferrari, xvith century), using only algebraic operations and radicals (nth roots). However, the case of degree 5 remained unsolved until Abel showed in 1826 that a general equation of degree 5 cannot be solved that way.
Soon after that, Galois defined the group of a polynomial equation as the group of permutations of its roots (say, complex roots) that preserve all algebraic identities with rational coefficients satisfied by these roots. Examples of such identities are given by the elementary symmetric polynomials, for it is well known that the coefficients of a polynomial are (up to sign) elementary symmetric polynomials in the roots. In general, all relations are obtained by combining these, but sometimes there are new ones and the group of the equation is smaller than the whole permutation group.
You will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. For example, it has been known for centuries how to explicitely solve polynomial equations of degree 2 (Babylonians, many centuries ago), 3 (Scipione del Ferro, Tartaglia, Cardan, around 1500 a.d.), and even 4 (Cardan, Ferrari, xvith century), using only algebraic operations and radicals (nth roots). However, the case of degree 5 remained unsolved until Abel showed in 1826 that a general equation of degree 5 cannot be solved that way.
Soon after that, Galois defined the group of a polynomial equation as the group of permutations of its roots (say, complex roots) that preserve all algebraic identities with rational coefficients satisfied by these roots. Examples of such identities are given by the elementary symmetric polynomials, for it is well known that the coefficients of a polynomial are (up to sign) elementary symmetric polynomials in the roots. In general, all relations are obtained by combining these, but sometimes there are new ones and the group of the equation is smaller than the whole permutation group.
Galois understood how this symmetry group can be used to characterize
the solvability of the equation. He defined the notion of solvable group and
showed that if the group of the equation is solvable, then one can express its
roots with radicals, and conversely.
Telling this story will lead us along interesting paths. You will, for example, learn why certain problems of construction by ruler and compass which were posed by the ancient Greeks and remained unsolved for centuries have no solution. On the other hand, you will know why (and maybe discover how) one can construct certain regular polygons.
There is an analogous theory for linear differential equations, and we will introduce a similar group of matrices. You will also learn why the explicit computation of certain indefinite integrals, such as integral of exp(x*x), is hopeless.
Telling this story will lead us along interesting paths. You will, for example, learn why certain problems of construction by ruler and compass which were posed by the ancient Greeks and remained unsolved for centuries have no solution. On the other hand, you will know why (and maybe discover how) one can construct certain regular polygons.
There is an analogous theory for linear differential equations, and we will introduce a similar group of matrices. You will also learn why the explicit computation of certain indefinite integrals, such as integral of exp(x*x), is hopeless.
On the menu are also some theorems from analysis: the transcendance
of the number π, the fact that the complex numbers form an algebraically
closed field, and also Puiseux’s theorem that shows how one can parametrize
the roots of polynomial equations, the coefficients of which are allowed to
vary.
There are some exercices at the end of each chapter. Please take some time to look at them. There is no better way to feel at ease with the topics in this book. Don’t worry, some of them are even easy!
There are some exercices at the end of each chapter. Please take some time to look at them. There is no better way to feel at ease with the topics in this book. Don’t worry, some of them are even easy!
Title : A field guide to algebra
author(s) : Antoine Chambert-Loir
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