## Abstract Algebra with Applications: Volume 1: Vector Spaces and GroupsA comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains. |

### Contents

VECTOR SPACES | 1 |

Examples from geometry 1 Vectors as equivalence classes of arrows 2 Addition | 19 |

Examples from electrical engineering and economics 22 Fields 24 Matrices 25 | 42 |

Vector spaces | 49 |

Polynomial and power series rings | 54 |

Vector spaces and subspaces 49 Examples 50 Free and generating sets 54 Bases | 55 |

Dimension 58 Dimension formula for subspaces 58 Direct decompositions 59 Quo | 65 |

Polynomials in one or more variables 54 Division with remainder 57 Roots and their | 67 |

Pure inseparability 275 Decomposition of an algebraic field extension into a separable | 278 |

Field theory and integral ring extensions | 291 |

Scalar products 273 Orthogonality 276 Orthogonal decompositions 279 Orthonor | 296 |

Affine algebras 301 Noethers normalization theorem 302 Zariskis lemma 304 | 304 |

Ring theory and algebraic geometry | 318 |

Adjoint of a linear mapping 303 Properties 304 Trace and operator norm 306 Nor | 324 |

Bilinear forms | 331 |

Affine varieties 318 Correspondence between varieties and ideals 322 Hilberts Null | 342 |

Linear and affine mappings | 72 |

Ideals 76 Ideals generated by a set 77 Principal ideals 77 Simple rings 78 Quotient | 86 |

phism theorem 82 Dimension formula for linear mappings 83 Vector spaces of linear | 88 |

Abstract affine geometry | 95 |

Affine spaces and subspaces 95 Dimension formula for affine subspaces 99 Affine | 105 |

Primary decompositions 102 Symbolic powers 105 Exercises | 107 |

Representation of linear mappings by matrices | 111 |

Parameterizations basis isomorphisms and coordinate transformations 111 Matrix | 119 |

principal ideal domains 119 Euclidean domains 119 Examples 120 Euclidean algo | 128 |

to equivalence 122 Similarity of square matrices 124 Canonical forms for projections | 130 |

Determinants | 138 |

Determinants as natural constructions in elimination theory 138 Existence and unique | 154 |

Transmission of the unique factorization property from R to Rx1 xn 135 Unique | 155 |

product by minors of the individual factors 165 Determinant of an endomorphism as | 173 |

Quadratic reciprocity law 166 Three theorems of Fermat 169 RamanujanNagell the | 178 |

Gershgorins theorem 183 Geometric and algebraic multiplicity 185 Diagonalizability | 186 |

Modules 185 Free modules 185 Submodules and quotient modules 186 Module | 196 |

Classification of endomorphisms up to similarity | 205 |

Cohens theorem 205 Noetherian induction 208 Primary decompositions in Noethe | 215 |

Field extensions | 216 |

Determinantal and elementary divisors 219 Exercises | 223 |

Characterization of simple extensions 227 Algebraically generating and independent | 238 |

Complexification 230 Tensor product of vector spaces 231 Properties 232 Basefield | 245 |

Adjoining roots of polynomials 248 Splitting fields 250 Normal extensions 250 | 262 |

Oriented areas and volumes 255 Vector product 258 Applications in geometry | 268 |

of alternating forms 341 Pfaffian 342 Classification of real symmetric forms by rank | 358 |

Tangent space of a variety at a point 348 Regular and singular points 351 Coinci | 361 |

Factorization of ideals | 365 |

Solving polynomial equations | 379 |

Grouptheoretical interpretation of the Gaussian algorithm 373 Gaussian decompo | 384 |

Markov processes 393 Transition matrices and graphs 393 Examples 394 Longterm | 400 |

Galois group of a field extension 388 Examples 388 Fixed fields 390 Correspondence | 401 |

405 Cyclic classes 407 Mean first passage matrix 411 Examples in genetics 413 | 413 |

Galois groups of finite fields 413 Primitive element theorem 414 Constructions with | 429 |

Examples from electrical engineering and economics 424 Existence and uniqueness | 444 |

Galois groups of cubic and quartic polynomials 443 Galois groups of certain polyno | 450 |

Symmetries in art and nature 451 Symmetry transformations 453 Types of one | 460 |

Roots of unity 455 Cyclotomic polynomials 457 Irreducibility of the cyclotomic poly | 463 |

Subgroups and cosets | 476 |

Lagranges theorem 487 Applications in number theory 488 Exercises | 489 |

Cycle decomposition 498 Cyclestructure and conjugacy 498 Generating sets | 505 |

522 Exercises | 523 |

groups 536 Semidirect products 537 Exercises | 541 |

Free groups 549 Universal property 550 Generators and relations 553 Von Dycks | 557 |

Torsion subgroup of an abelian group 557 Classification of finitely generated abelian | 570 |

Group actions 575 Examples 575 Orbits and stabilizers 577 Transitivity 577 | 591 |

Actions of groups on groups and coset spaces 596 Class equation 597 Deriving struc | 604 |

of finite nilpotent groups 617 Affine subgroups of Sym 617 Characterization of | 622 |

Topological groups and their subgroups 624 Continuous homomorphisms 628 Iso | 641 |

### Common terms and phrases

a₁ abelian group affine space affine subspace algebraic arbitrary automorphism b₁ base-field basis v1 called canonical characteristic polynomial claim coefficients column commutes conditions are equivalent Consider decomposition defined definition denote determinant diagonal matrix diagonalizable eigenvalues eigenvector elements endomorphism equation Euclidean space exactly example fact field finite-dimensional vector space following conditions geometric given GL(n group G hence Hint homomorphism implies invertible matrix isomorphism K-vector Kmxn Kn×n Knxn Let G linear mapping linearly independent matrix representation Moreover n-dimensional natural number nilpotent nonzero normal subgroup obtained orthogonal orthonormal basis permutation plane Problem Proof Proposition prove rank real numbers real vector space respect satisfies scalar multiplication scalar product semisimple Show solution space and let span subgroup of G subset Suppose surjective theorem topological groups topology U₁ U₂ unique v₁ volume function zero