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Field Theory

Steven Roman
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics
[Reviewed by
Darren Glass
, on

Springer has just released the second edition of Steven Roman's Field Theory, and it continues to be one of the best graduate-level introductions to the subject out there. The book has been significantly rewritten, based on comments from readers of the first edition as well as repeated testing of the book in his own graduate classes, according to the author's preface.

The book opens with a 0th chapter of preliminaries, giving a (very) quick recap of the theories of lattices, groups, rings, domains, quotient fields, and tensor products among other topics — all in under twenty pages. The bulk of the book is divided into three parts, the first of which is devoted to field extensions. Roman begins this part by discussing properties of polynomials defined over rings and fields, before moving into a lengthy discussion of different types of field extensions, embeddings, separability and independence

The second part of the book deals with Galois theory, and opens with a very nicely written chapter treating the history of Galois theory from the Babylonian methods of solving quadratic equations through Newton's work on symmetric polynomials to Galois's deep insights. Vandermonde and Lagrange both play large roles in this story, and this chapter is an excellent combination of history and mathematics. The book then moves deeper into the theory, with chapters on the Galois correspondence between subextensions of a subfield and subgroups of the Galois group, and the Galois groups associated to polynomials. There is a particularly nice discussion in these chapters of the Fundamental Theorem of Algebra, and a proof is given which uses Galois's ideas.

The third part of the book is devoted to binomials — that is, polynomials of the form xn – u. The first chapter gives a hint as to the complexity of these innocent-looking polynomials, as Roman considers just the case where u = 1, and looks at the complexity of these so-called cyclotomic extensions. This chapter ends with a brief discussion of the Inverse Galois problem, which asks what finite groups can be realized as Galois groups over the field of rational numbers. The next chapters continue this discussion of binomials, with chapters dedicated to cyclic extensions and solvable extensions, before moving on to more general situations and a brief introduction to Kummer theory.

Every section of the book has a number of good exercises that would make this book excellent to use either as a textbook or to learn the material on your own. All in all, I recommend this book highly as it is a well-written expository account of a very exciting area in mathematics.

Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include algebraic geometry, number theory, and (wait for it) Galois Theory. He can be reached at

Preface to the Second Edition, vii

Preface to the First Edition, ix

Preliminaries, 1

Part 1 Preliminaries, 1

Part 2 Algebraic Structures, 16

Part I—Basic Linear Algebra, 31

1 Vector Spaces, 33

Vector Spaces, 33

Subspaces, 35

Direct Sums, 38

Spanning Sets and Linear Independence, 41

The Dimension of a Vector Space, 44

Ordered Bases and Coordinate Matrices, 47

The Row and Column Spaces of a Matrix, 48

The Complexification of a Real Vector Space, 49

Exercises, 51

2 Linear Transformations, 55

Linear Transformations, 55

Isomorphisms, 57

The Kernel and Image of a Linear Transformation, 57

Linear Transformations from ���� to ����, 59

The Rank Plus Nullity Theorem, 59

Change of Basis Matrices, 60

The Matrix of a Linear Transformation, 61

Change of Bases for Linear Transformations, 63

Equivalence of Matrices, 64

Similarity of Matrices, 65

Similarity of Operators, 66

Invariant Subspaces and Reducing Pairs, 68

xii Contents

Topological Vector Spaces, 68

Linear Operators on �� ��, 71

Exercises, 72

3 The Isomorphism Theorems, 75

Quotient Spaces, 75

The Universal Property of Quotients and

the First Isomorphism Theorem, 77

Quotient Spaces, Complements and Codimension, 79

Additional Isomorphism Theorems, 80

Linear Functionals, 82

Dual Bases, 83

Reflexivity, 84

Annihilators, 86

Operator Adjoints, 88

Exercises, 90

4 Modules I: Basic Properties, 93

Modules, 93

Motivation, 93

Submodules, 95

Spanning Sets, 96

Linear Independence, 98

Torsion Elements, 99

Annihilators, 99

Free Modules, 99

Homomorphisms, 100

Quotient Modules, 101

The Correspondence and Isomorphism Theorems, 102

Direct Sums and Direct Summands, 102

Modules Are Not As Nice As Vector Spaces, 106

Exercises, 106

5 Modules II: Free and Noetherian Modules, 109

The Rank of a Free Module, 109

Free Modules and Epimorphisms, 114

Noetherian Modules, 115

The Hilbert Basis Theorem, 118

Exercises, 119

6 Modules over a Principal Ideal Domain, 121

Annihilators and Orders, 121

Cyclic Modules, 122

Free Modules over a Principal Ideal Domain, 123

Torsion-Free and Free Modules, 125

Contents xiii

Prelude to Decomposition: Cyclic Modules, 126

The First Decomposition, 127

A Look Ahead, 127

The Primary Decomposition, 128

The Cyclic Decomposition of a Primary Module, 130

The Primary Cyclic Decomposition Theorem, 134

The Invariant Factor Decomposition, 135

Exercises, 138

7 The Structure of a Linear Operator, 141

A Brief Review, 141

The Module Associated with a Linear Operator, 142

Orders and the Minimal Polynomial, 144

Cyclic Submodules and Cyclic Subspaces, 145

Summary, 147

The Decomposition of ���� , 147

The Rational Canonical Form, 148

Exercises, 151

8 Eigenvalues and Eigenvectors, 153

The Characteristic Polynomial of an Operator, 153

Eigenvalues and Eigenvectors, 155

Geometric and Algebraic Multiplicities, 157

The Jordan Canonical Form, 158

Triangularizability and Schur's Lemma, 160

Diagonalizable Operators, 165

Projections, 166

The Algebra of Projections, 167

Resolutions of the Identity, 170

Spectral Resolutions, 172

Projections and Invariance, 173

Exercises, 174

9 Real and Complex Inner Product Spaces , 181

Norm and Distance, 183

Isometries, 186

Orthogonality, 187

Orthogonal and Orthonormal Sets, 188

The Projection Theorem and Best Approximations, 192

Orthogonal Direct Sums, 194

The Riesz Representation Theorem, 195

Exercises, 196

10 Structure Theory for Normal Operators, 201

The Adjoint of a Linear Operator, 201

xiv Contents

Unitary Diagonalizability, 204

Normal Operators, 205

Special Types of Normal Operators, 207

Self-Adjoint Operators, 208

Unitary Operators and Isometries, 210

The Structure of Normal Operators, 215

Matrix Versions, 222

Orthogonal Projections, 223

Orthogonal Resolutions of the Identity, 226

The Spectral Theorem, 227

Spectral Resolutions and Functional Calculus, 228

Positive Operators, 230

The Polar Decomposition of an Operator, 232

Exercises, 234

Part II—Topics, 235

11 Metric Vector Spaces: The Theory of Bilinear Forms, 239

Symmetric, Skew-Symmetric and Alternate Forms, 239

The Matrix of a Bilinear Form, 242

Quadratic Forms, 244

Orthogonality, 245

Linear Functionals, 248

Orthogonal Complements and Orthogonal Direct Sums, 249

Isometries, 252

Hyperbolic Spaces, 253

Nonsingular Completions of a Subspace, 254

The Witt Theorems: A Preview, 256

The Classification Problem for Metric Vector Spaces, 257

Symplectic Geometry, 258

The Structure of Orthogonal Geometries: Orthogonal Bases, 264

The Classification of Orthogonal Geometries:

Canonical Forms, 266

The Orthogonal Group, 272

The Witt's Theorems for Orthogonal Geometries, 275

Maximal Hyperbolic Subspaces of an Orthogonal Geometry, 277

Exercises, 279

12 Metric Spaces, 283

The Definition, 283

Open and Closed Sets, 286

Convergence in a Metric Space, 287

The Closure of a Set, 288

Contents xv

Dense Subsets, 290

Continuity, 292

Completeness, 293

Isometries, 297

The Completion of a Metric Space, 298

Exercises, 303

13 Hilbert Spaces, 307

A Brief Review, 307

Hilbert Spaces, 308

Infinite Series, 312

An Approximation Problem, 313

Hilbert Bases, 317

Fourier Expansions, 318

A Characterization of Hilbert Bases, 328

Hilbert Dimension, 328

A Characterization of Hilbert Spaces, 329

The Riesz Representation Theorem, 331

Exercises, 334

14 Tensor Products, 337

Universality, 337

Bilinear Maps, 341

Tensor Products, 343

When Is a Tensor Product Zero? 348

Coordinate Matrices and Rank, 350

Characterizing Vectors in a Tensor Product, 354

Defining Linear Transformations on a Tensor Product, 355

The Tensor Product of Linear Transformations, 357

Change of Base Field, 359

Multilinear Maps and Iterated Tensor Products, 363

Tensor Spaces, 366

Special Multilinear Maps, 371

Graded Algebras, 372

The Symmetric Tensor Algebra, 374

The Antisymmetric Tensor Algebra:

The Exterior Product Space, 380

The Determinant, 387

Exercises, 391

15 Positive Solutions to Linear Systems:

Convexity and Separation 395

Convex, Closed and Compact Sets, 398

Convex Hulls, 399

xvi Contents

Linear and Affine Hyperplanes, 400

Separation, 402

Exercises, 407

16 Affine Geometry, 409

Affine Geometry, 409

Affine Combinations, 41

Affine Hulls, 412

The Lattice of Flats, 413

Affine Independence, 416

Affine Transformations, 417

Projective Geometry, 419

Exercises, 423

17 Operator Factorizations: QR and Singular Value, 425

The QR Decomposition, 425

Singular Values, 428

The Moore–Penrose Generalized Inverse, 430

Least Squares Approximation, 433

Exercises, 434

18 The Umbral Calculus, 437

Formal Power Series, 437

The Umbral Algebra, 439

Formal Power Series as Linear Operators, 443

Sheffer Sequences, 446

Examples of Sheffer Sequences, 454

Umbral Operators and Umbral Shifts, 456

Continuous Operators on the Umbral Algebra, 458

Operator Adjoints, 459

Umbral Operators and Automorphisms

of the Umbral Algebra, 460

Umbral Shifts and Derivations of the Umbral Algebra, 465

The Transfer Formulas, 470

A Final Remark, 471

Exercises, 472

References, 473

Index, 475