Fundamentals of Advanced Mathematics V2: Field extensions, topology and topological vector spaces, functional spaces, and sheaves

By Henri Bourles

The three volumes of this series of books, of which this is the second, put forward the mathematical elements that make up the foundations of a number of contemporary scientific methods: modern theory on systems, physics and engineering.

Whereas the first volume focused on the formal conditions for systems of linear equations (in particular of linear differential equations) to have solutions, this book presents the approaches to finding solutions to polynomial equations and to systems of linear differential equations with varying coefficients.

Fundamentals of Advanced Mathematics, Volume 2: Field Extensions, Topology and Topological Vector Spaces, Functional Spaces, and Sheaves begins with the classical Galois theory and the theory of transcendental field extensions. Next, the differential side of these theories is treated, including the differential Galois theory (Picard-Vessiot theory of systems of linear differential equations with time-varying coefficients) and differentially transcendental field extensions. The treatment of analysis includes topology (using both filters and nets), topological vector spaces (using the notion of disked space, which simplifies the theory of duality), and the radon measure (assuming that the usual theory of measure and integration is known).

In addition, the theory of sheaves is developed with application to the theory of distributions and the theory of hyperfunctions (assuming that the usual theory of functions of the complex variable is known). This volume is the prerequisite to the study of linear systems with time-varying coefficients from the point-of-view of algebraic analysis and the algebraic theory of nonlinear systems.

• Present Galois Theory, transcendental field extensions, and Picard
• Includes sections on Vessiot theory, differentially transcendental field extensions, topology, topological vector spaces, Radon measure, differential calculus in Banach spaces, sheaves, distributions, hyperfunctions, algebraic analysis, and local analysis of systems of linear differential equations

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 3, 2018
• ISBN: 9780081023853

Henri Bourles

Henri Bourlès is Full Professor and Chair at the Conservatoire National des Arts et Métiers, Paris, France.

Book preview

Fundamentals of Advanced Mathematics 2

Field Extensions, Topology and Topological Vector Spaces, Functional Spaces, and Sheaves

Henri Bourlès

New Mathematical Methods, Systems and Applications Set

coordinated by

Henri Bourlès

Table of Contents

Cover

Title page

Copyright

Preface

Errata for Volume 1

List of Notation

Chapter 1: Field Extensions and Differential Field extensions

Chapter 2: General Topology

Chapter 3: Topological Vector Spaces

Chapter 4: Measure, Integration, Function spaces

Chapter 5: Sheaves

1: Field Extensions and Differential Field Extensions

Abstract

1.1 Galois theory

1.2 Transcendental extensions

1.3 Differential Galois theory

1.4 Differentially transcendental extensions

2: General Topology

Abstract

2.1 Introduction to general topology

2.2 Filters and nets

2.3 Topological structures

2.4 Uniform structures

2.5 Bornologies

2.6 Baire spaces, Polish spaces, Suslin spaces, and Lindelöf spaces

2.7 Uniform function spaces

2.8 Topological algebra

3: Topological Vector Spaces

Abstract

3.1 Introduction

3.2 General topological vector spaces

3.3 Locally convex spaces

3.4 Important types of locally convex spaces

3.5 Weak topologies

3.6 Dual of a locally convex space

3.7 Bidual and reflexivity

-spaces and their duals

3.9 Continuous multilinear mappings

3.10 Hilbert spaces

3.11 Nuclear spaces

4: Measure and Integration, Function Spaces

Abstract

4.1 Measure and integration

4.2 Functions in a single complex variable

4.3 Function spaces

4.4 Generalized function spaces

5: Sheaves

Abstract

5.1 Introduction

5.2 General results about sheaves

5.3 Sheaves of Modules

5.4 Cohomology of sheaves

Bibliography

Cited Authors

Index

Copyright

First published 2018 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

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Preface

Henri Bourlès October 2017

The first volume [P1] of this Précis gave conditions for solving systems of equations; this included some polynomial equations and a few short detours into algebraic geometry, but we were first and foremost interested in linear differential equations. With the exception of the elementary case of constant coefficients, we arrived at conditions that were admittedly still formal, all too formal, just as Nietzsche once wrote human, all too human. The goal of this second volume is to add flesh to the bones of this mathematical skeleton. One approach to finding solutions to equations with coefficients in the field K is to look in the field extension L/K. In the case of polynomial equations, this leads us to Galois theory, and, for differential equations, to differential Galois theory (Picard-Vessiot theory, whose most decisive results were established by E. Kolchin); there is an exact parallel between these two theories. Galois theory is presented in detail in Chapter 1, including a full proof of the Abel-Ruffini theorem, showing that the general quintic equation cannot be solved by radicals, a question that tortured mathematicians for three centuries. The main results of differential Galois theory demand long and difficult proofs, and so we shall typically omit them, as each result is somewhat justified by its counterpart in Galois theory.

Once we have vanquished this deeply algebraic first chapter, we can postpone the study of analysis no longer. More precisely, we shall focus on functional analysis, where the class of functions should be understood to include generalized functions in the sense of I. Gelfand [GEL 68] (essentially Schwartz distributions), or even some more exotic objects such as Sato’s hyperfunctions. The latter will in fact provide us with the analytic tools needed to solve systems of differential equations with variable coefficients, as we shall see in section 5.4.6 at the end of this volume.

But before we arrive at this lofty goal, we must first lay the foundations of analysis, beginning with topology, which is presented in Chapter 2. We shall assume that elementary topological concepts in metric spaces are known. These concepts can be found in references such as the Foundations of Modern Analysis by J. Dieudonné, (1st volume of the Elements of Analysis [DIE 82]). Since most distribution spaces are not metrizable, we will need to study general topological spaces and their variants (uniform spaces, etc.). In metric spaces, sequences are a valuable instrument for proving all kinds of theorems and results; but in non-metrizable spaces we can no longer rely on them, and must replace them with generalized sequences such as the nets proposed by Moore-Smith, or the filters of H. Cartan. We will study these concepts at the start of Chapter 2, showing how nets and filters are equivalent in a strictly logical sense, but complementary on a psychological level. In section 2.5, it seemed useful to introduce the notion of bornology on a general Lipschitz space rather than just a locally convex topological vector space, in particular because the notion of bounded set in a metric space (without any vector-space structure) is familiar. It will be very convenient to have one single general framework. We used quantifiers, especially in topology, in an attempt to lighten the text (breaking with Bourbakian and other conventions), and we added more parenthesis and commas than the logicians would conventionally recommend, seeking clarity above all else.

Generalized function spaces (let us denote one such space by E′ for now) may be constructed by taking the dual of a test function space E. Suppose that E is the L² space of square-integrable functions on the real line. Since E is a Hilbert space (Theorem 4.12), it is self-dual (Theorem 3.151). Now, if we choose a test function space E that is smaller than L², its dual E′ will be larger, and the smaller we make E, the larger Eis much smaller than L, the space of distributions, is much larger.

Topological vector spaces, which include the theory of dual spaces, are studied in Chapter 3. The most important such spaces for applications are the locally convex spaces. The theory of dual spaces may be developed naturally on these spaces; we shall therefore focus on them for the majority of our presentation. The two most classic types of locally convex space are the Banach spaces (the complete normed vector spaces) and the Hilbert spaces, a special case of the former. We mentioned earlier that L² is a Hilbert space. We will see that the spaces Lp (p ∈ [1, ∞]) are in fact Banach spaces. But many of the spaces that we wish to study here are neither, and so we cannot restrict ourselves to Banach and Hilbert spaces alone. Fréchet spaces and their inductive limits, Montel spaces, and Schwartz spaces (section 3.4) will all have key roles to play; the latter two types of space are always non-metrizable except in finite dimensions. Furthermore, we will constantly require weak topologies, for instance in distribution theory, and so we shall study them in detail in section 3.5. Reflexivity is also studied in section 3.7: given a space E, is E canonically isomorphic to its bidual (the dual of its dual when both are given the strong topology)? If E is a finite-dimensional space, we already know that the answer is yes ([P1], section 3.1.3(VI)). We will see that Lp is reflexive for each p ∈ ]1, ∞ [, but is not reflexive in general for p = 1 or p = ∞ (Corollary 4.17 and Remark 4.18). The limited scope of this volume sadly did not allow the inclusion of the elegant theory of compact operators developed by F. Riesz, and related questions (Fredholm theory, Hilbert-Schmidt operators, Sturm-Liouville problem, etc.), but interested readers can find a comprehensive presentation in the Foundations of Modern Analysis. between the distribution space on X × Y (where X and Y -valued distributions on Yare both nuclear spaces (section 4.3.2(III), Remark 4.84), and has deep connections with Fredholm theory [GRO 56].

Armed with our discoveries from Chapters 2 and 3, we will finally be able to study generalized function spaces in Chapter 4. Our first rendez-vous is measure and integration theory, a truly fabulous tool crafted by E. Borel, H. Lebesgue, and their successors without which none of the rest would be possible. We will encounter our first type of generalized function: Radon measures. There are two separate theories of integration: one founded on abstract measures, the other founded on Radon measures. Bourbaki [BKI 69] adopted the second approach, for entirely defensible reasons, and was strongly criticized for doing so, for equally defensible reasons, most notably due to the significant difficulties involved in constructing any reasonable theory of probability using Radon measures ([BKI 69], Chapter IX), [SCW 73]. In section 4.1, rather than choosing sides and restricting ourselves to either one of two approaches, we shall instead examine the parallels between them and allow ourselves the luxury of selecting the most appropriate in any given case. Although the theory of functions in a single complex variable is covered by the Foundations of Modern Analysis (and is deeply embedded within engineering culture), we shall reiterate it here in section 4.2 using the concept of homology (following the approach initiated by Ahlfors [AHL 66]), rather than the concept of homotopy employed by the Foundations. As well as being more general ([P1], section 3.3.8(II)), the notion of homology is more geometric, and ultimately more convenient. The concept of a meromorphic function will be presented (section 4.2.6) using the Mittag-Leffler and Weierstrass theorems (whose proofs are omitted) to prepare for the subsequent chapter. We then move on to the classic function spaces (spaces of infinitely differentiable and analytic functions) and their duals (distributions and hyperfunctions). Distribution theory can be viewed as the culmination of the theory of topological vector spaces, whose developement spanned from Hilbert until the 1950s. M. Sato had the idea to use analytic functions as test functions in order to construct hyperfunctions; however, there are no non-zero analytic functions with compact support, which prevented the theory of hyperfunctions from being a more or less direct application of the classic theory of duality. Hyperfunctions are the boundary values of holomorphic functions, or alternatively cohomology spaces whose values belong to a sheaf of holomorphic functions. Hence, although it is relatively simple to define hyperfunctions in a single variable (section 4.4.2), we can only understand their deeper nature or attempt to generalize them to multiple variables once we have studied sheaf theory.

The latter, developed by J. Leray and later H. Cartan, is therefore the subject of our final chapter (Chapter 5); we will focus on sheaves of Modules on ringed spaces (section 5.3) and their cohomology (section 5.4). Coherent algebraic sheaves and coherent analytic sheaves, the vital ingredients of algebraic geometry and analytic geometry respectively (in the sense of J.P. Serre for the latter), will enjoy a privileged status in our considerations. We will consider two applications: meromorphic functions in several variables (section 5.4.4) and hyperfunctions (section 5.4.5). Finally, we will see how the latter may be applied to systems of differential equations with variable coefficients to obtain a result that is analogous (despite requiring computations that are effectively much more difficult) to the result obtained in ([P1], section 3.4.4) for systems of differential equations with constant coefficients; this final section will adopt a heuristic approach, aiming to gradually narrow down the possibilities.

Errata for Volume 1

p. IX, lines 10-11: omit together with… [MAC 14]

p. 24, lines 4 and 5: read inductive instead of injective

p. 54, line 9: read (III),(IV) instead of (IV),(V)

p. 63, line 22: read of elements of instead of ∈

p. 68, line 11: after is, read thus. After if, read and only if

p. 71, line 1: delete domain

p. 148, line 12: read induced by a function instead of one

p. 165, lines 7 and 9: read HomB instead of HomB

p. 180, line 14: read A instead of A

p. 216, line 4: read K instead of K

List of Notation

Chapter 1: Field Extensions and Differential Field extensions

(f) : set of roots of the polynomial f, p. 2

Gal (f): Galois group of the polynomial f, p. 2

, of {1, …, n}, p. 2

[L : K] := dimK (L), p. 3

L/K : field extension, p. 4

K (α), K (E), p. 4

: algebraic closure of K, p. 4

: field of algebraic numbers, p. 4

Gal (L/K) : Galois group of the Galois extension L/K, p. 6

M┴ = Gal (N/M), p. 6

Δ┴ : fixed field of the group Δ, p. 6

, p. 6

K (X1, …, Xn) : field of rational fractions in X1, …, Xn, p. 8

si (i = 1, …, n) : elementary symmetric polynomials, p. 8

~≡: not congruent to, p. 10

Σ (f, g) : Sylvester matrix of the polynomials f and g, p. 15

Res (f, g) : resultant of the polynomials f and g, p. 15

Δ (f) : discriminant of the polynomial f, p. 17

GF(pn) : Galois field, p. 18

μh : group of h-th roots of unity, p. 19

φ (h) : Euler’s totient function evaluated at h, p. 19

[L : K]s : separable degree of the algebraic extension L/K, p. 22

: separable closure of K, p. 24

(i) : field of Gaussian rationals, p. 27

[i] : ring of Gaussian integers, p. 27

deg trKE : transcendence degree of the extension E over K, p. 39

K 〈v1, …, vm〉 : differential field extension, p. 40

W (u1, …, un) : Wronksian, p. 42

GalD (M/K) : differential Galois group, p. 44

Dn (C) : group of invertible diagonal matrices, p. 45

Gm (C) : multiplicative group of C×, p. 45

Tn (C) : group of invertible upper triangular matrices, p. 45

Un (C) : group of unipotent upper triangular matrices, p. 45

Ga (C) : additive group of C, p. 45

On (S; C), On (S; C) : orthogonal group, p. 45

SOn (C) : special orthogonal group, p. 45

G° : connected component of the neutral element of the algebraic group G, p. 46

∫ v : primitive, p. 48

e∫ v: exponential of a primitive, p. 49

deg trD (F/K): differential transcendence degree, p. 53

Chapter 2: General Topology

B (a; r), resp. Bc (a; r): open ball, resp. closed ball, p. 55

δ (A): diameter of A, p. 56

: image under f , p. 62

: image under f , p. 65

: closure of A, p. 65

: interior of A, p. 65

Fr (A) = ∂ (A): frontier of A, p. 65

: set of open neighborhoods of A, p. 65

: filter of neighborhoods of a, p. 67

: convergence to a, p. 67

, p. 69

, f(a + 0), f(b − 0), p. 69

, p. 69

Gr(f), Gr(~): graph, p. 71

χA : characteristic function of A, p. 71

X∞ : one-point compactification of the locally compact space X, p. 80

A ⋐ B, p. 80

supp (f): support of the numerical function f, p. 83

W ∘ W′ : p. 84

Ud (r), Udc (r): p. 84, 88

Topu : category of uniform spaces, p. 87

Bd (x; r), Bdc (x; r): open semi-ball, closed semi-ball, p. 88

[d]: Lipschitz equivalence class of the pseudometric d, p. 90

Topl : category of Lipschitz spaces, p. 90

: Hausdorff completion of the uniform space X, p. 93

Topuc : category of uniform Hausdorff complete spaces, p. 93

Bor : category of bornological sets, p. 98

: bornology, p. 99

s : discrete bornology, p. 99

c : bornology of compact subsets, p. 99

pc : bornology of precompact subsets, p. 99

b : canonical bornology, p. 99

u : trivial bornology, p. 99

: set of mappings from X into Y, p. 103

: p. 103

: set of continuous mappings from X into Y.

: p. 105

: set of bounded mappings from E into F, p. 109

: equibornology, p. 109

Topgrp : category of topological groups, p. 109

: left, right uniform structure of a topological group, p. 110

ker (u), im (u), coim (u): kernel, image, coimage, p. 113

Topab: category of abelian topological groups, p. 115

coker (u): cokernel, p. 115

Chapter 3: Topological Vector Spaces

: field of real or complex numbers, p. 118

, |·| : p. 118, 121

Vec -vector spaces, p. 118

denotes the topology on E), p. 118

Σi ∈ Ixi : (possibly infinite) sum of terms, p. 119

Tvs : category of topological vector spaces, p. 120

: space of continuous linear mappings from E into F, p. 120

Hom (E; F): space of linear mappings from E into F, p. 120

E* : algebraic dual, p. 120

E′ : dual, p. 120

: ring of continuous endomorphisms of E, p. 120

Tvsh : category of Hausdorff topological vector spaces, p. 120

, p. 120

[A]: vector space generated by A, p. 120.

codim (M): codimension of M, p. 123

[x, y], ]x, y[ : closed interval, open interval, p. 126

epi (f): epigraph of f, p. 126

Bp (α), Bpc (α), p. 127

pA : gauge of the absorbing disc A, p. 128

Lcs : category of locally convex spaces, p. 133

Lcsh : category of locally convex Hausdorff spaces, p. 133

⊕i ∈ IEi : topological direct sum, p. 134

: p. 136

, p. 139

|| u || : norm of the continuous linear mapping u, p. 139

: p. 141

EV : normed vector space associated with V, p. 142

: completion of EV, p. 142

Bpc (α; F) : p. 144

: space of compact operators from E into F, p. 148

: p. 150

σ (E, F): weak topology of E, p. 151

tu : transpose of u, p. 153

Es′, Es : weak* dual, space E equipped with the weak topology, p. 157

τ (E, F): Mackey topology, p. 158

Eτ, Eτ′ : spaces E, E′ equipped with the Mackey topology, p. 158

β (E′, E), Eb′ : strong topology, strong dual, p. 159

-topology, p. 159

E″, cE : bidual, canonical mapping E → E″, p. 163

γ (E′, E): topology of compact convergence of the dual, p. 168

|| u || : norm of the multilinear mapping u, p. 177

, p. 177

〈x | y〉 : scalar product, p. 178

A⊥ : orthogonal of A, p. 179

pH (x): orthogonal projection of x onto H, p. 180

σΕ : anti-linear bijection E′ → E, p. 184

: Hilbertian sum, p. 185

u* : adjoint of the operator u, p. 188

: space of Hermitian operators in E, p. 188

U (E), O (E) : group of unitary endomorphisms, of symmetric endomorphisms, p. 188

: cone of positive operators in E, p. 188

Chapter 4: Measure, Integration, Function spaces

: set of positive, real, complex, signed measures, p. 199

λ , p. 200

δa : Dirac measure or distribution at the point a, p. 200, 213, 245

ϒ: Heaviside function, p. 200

: completion of the measure μ, p. 200

~μ : equivalence mod .μ, p. 201

: equivalence class of f (mod .μ), p. 201

∫ f. dμ, ∫Xf(x). dμ(x) : Bochner integral of f, p. 202

ϖ : Dirac comb, p. 202, 213, 246

ess sup, ess inf : essential supremum, essential infimum, p. 204

, Lp(X, μ; F), p. 204

Np : norm on Lp (X, μ; F), p. 204

, μ1 ⊗ μ2 : tensor product of σ-algebras, of measures, p. 207

λn, p. 207

, p. 209

: space of Radon measures, p. 210

supp (μ), supp (T): support of a measure, of a distribution, p. 211, 243

, N1,K (f, p. 211

μ+, μ−, | μ | : p. 213

| v | ≪ μ, p. 216

: Radon-Nikodym derivative, p. 216

h ↓ 0, p. 218

NBV ([a, b]) : space of functions of bounded variation, p. 219

μ ⊥ ν : disjoint measures, p. 221

AC ([a, b]) : space of absolutely continuous functions, p. 222

∫ZHzdπ(z) : continuous sum of Hilbert spaces, p. 223

: space of holomorphic functions, p. 225

∫γf(z). dz, ∮γf(z). dz : integral along a path, along a closed path, p. 225

: space of meromorphic functions, p. 233

, p. 232

: space of analytic functions with values in E, p. 239

α! := α1!…αn!, rα, p. 240

, p. 235

, p. 236

, p. 237

Cω : class of analytic functions, p. 239

: space of germs of holomorphic functions, of analytic functions, p. 242

: distribution spaces, p. 243

: space of infinitely differentiable slowly growing functions, p. 247

sing supp : singular support, p. 247

: space of hyperfunctions on S, p. 250

[f]z = x, f (x ± i0), p. 251

with compact support, p. 253

Chapter 5: Sheaves

ρVU : restriction morphism, p. 260

: category of open sets in B, p. 258

s |V : restriction of the section s, p. 260

: object of the sections over U, p. 260

: morphism of presheaves, p. 260

: category of presheaves of base B with values in C, p. 260

: induced presheaf on B′ ⊂ B, p. 260

, p. 260

under ψ, p. 260

V ≽ U (i.e. V ⊂ U), p. 261

, fb at the point b, morphism of stalks, p. 261

sb : germ of the section s at the point b, p. 261

: category of sheaves of base B with values in C, p. 263

, p. 264

: sheafification functor, p. 264

Γ: section functor, p. 264

, p. 266

Xb : fiber of a fibered space (X, B, p) at the point b ∈ B, p. 267

: sheaf of continuous sections of p, p. 267

, p. 269

: sheaf of mappings, of continuous mappings, p. 270

: sheaf of germs of homomorphisms, p. 271

: quotient sheaf, p. 275

-modules, p. 276

, p. 276

, p. 276

: canonical flabby resolution, p. 287

-modules, p. 280

-modules, p. 280

, p. 290

: sheaf of germs of holomorphic functions, p. 292

: sheaf of germs of meromorphic functions, p. 292

: sheaf of germs of divisors, p. 295

ordt (aj): order of the zero aj at the point t, p. 301

σi : greatest slope of the Newton polygon, p. 301

, p. 301

1

Field Extensions and Differential Field Extensions

Abstract

(I) The problem of solving algebraic equations by radicals (where by algebraic equations we mean those of the form f (x) = 0, for f a non-zero polynomial with rational coefficients) was one of the most important topics in mathematics from the earliest periods of antiquity until the 19th century. The Babylonians, and later the Greeks, already knew how to solve quadratic equations, although their formulas were more complex than those we use today, since their notation was inferior and they lacked the concepts of zero and negative numbers. Cubic equations were solved by S. del Ferro (around 1515) and N. Tartaglia (whose contributions were published in 1545 by G. Cardano and are often incorrectly attributed to the latter); quartic equations were solved by L. Ferrari using a method that was also published in Cardano’s Ars Magna together with the method proposed by Tartaglia. In 1576, R. Bombelli compiled a summary of all of this work using simpler notation in Algebra. However, all efforts to solve quintic equations were unsuccessful from the end of the 16th century until the early 19th century. J.-L. Lagrange was arguably the first to recognize the underlying reasons in his memoirs from 1770–1771. Finally, P. Ruffini attempted to show that the general quintic equation cannot be solved by radicals in a series of dense and controversial memoirs gradually published between 1799 and 1813. One proof that was entirely correct but weighed down by long calculations was provided by N. Abel in 1824; in 1826, he showed.

Keywords

Algebraic dependence; Decomposition fields; Field Extensions; Radicals; Simple adjunctions; Transcendence degree; Transcendental extensions

1.1 Galois theory

1.1.1 Introduction

(I) The problem of solving algebraic equations by radicals (where by algebraic equations we mean those of the form f (x) = 0, for f a non-zero polynomial with rational coefficients) was one of the most important topics in mathematics from the earliest periods of antiquity until the 19th century. The Babylonians, and later the Greeks, already knew how to solve quadratic equations, although their formulas were more complex than those we use today, since their notation was inferior and they lacked the concepts of zero and negative numbers. Cubic equations were solved by S. del Ferro (around 1515) and N. Tartaglia (whose contributions were published in 1545 by G. Cardano and are often incorrectly attributed to the latter); quartic equations were solved by L. Ferrari using a method that was also published in Cardano’s Ars Magna together with the method proposed by Tartaglia. In 1576, R. Bombelli compiled a summary of all of this work using simpler notation in Algebra¹. However, all efforts to solve quintic equations were unsuccessful from the end of the 16th century until the early 19th century. J.-L. Lagrange was arguably the first to recognize the underlying reasons in his memoirs from 1770-1771. Finally, P. Ruffini attempted to show that the general quintic equation cannot be solved by radicals in a series of dense and controversial memoirs gradually published between 1799 and 1813. One proof that was entirely correct but weighed down by long calculations was provided by N. Abel in 1824; in 1826, he showed:

Theorem 1.1

Abel-Ruffini

The general algebraic equation of degree n with rational coefficients may be solved by radicals if and only if n ≤ 4.

(II) É. Galois discovered the deeper reasons underlying this theorem by deriving it from group theory after defining the notion of a normal subgroup ([P1], section 2.2.2(I)be the set of n roots of a non-zero polynomial f. Assume that there are no duplicate roots. We shall begin by giving a definition of the Galois group of f and stating Galois’ theorem. Later, with the benefit of hindsight, this definition and this theorem will both reveal themselves to be provisional in nature (see below, Definition 1.10 and Theorem 1.15).

Definition 1.2

The Galois group of f is the subgroup Gal (f) of the symmetric group consisting of the permutations that fix every rational expression of the roots of f.

Theorem 1.3

Galois

The algebraic equation f (x) = 0, where is an irreducible polynomial of degree n > 0, may be solved by radicals if and only if Gal (f) is solvable ([P1], section 2.2.7(I)).

These new ideas and Galois’ theorem were presented in a memoir submitted to the Académie des Sciences de Paris in 1830, which was given to A.-L. Cauchy, who lost it. They were later retranscribed in the documents attached to the letter written by Galois the night before he died (May 31st, 1832). These documents were published in 1846 thanks to the efforts of J. Liouville had already been studied in detail by Lagrange (without any special focus on those which fix rational expressions), followed by Cauchy almost half a century later. The notion of solvable group was introduced by C. Jordan (1867) and its definitive form was established by O. Hölder (1889) in the theorem which bears both their names ([P1], section 2.2.5(II), Theorem 2.15).

(III) The scope and ramifications of Galois’ work extended much further than solving algebraic equations. His original manuscripts, so dense that they are difficult to decipher, were gradually pieced together throughout the second half of the 19th century by various authors, most notably E. Betti, J. Serret, and of course C.