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# Modern Algebra MATH 631

UWM

GPA 3.64

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This 14 page Class Notes was uploaded by Destiny Heaney on Tuesday October 27, 2015. The Class Notes belongs to MATH 631 at University of Wisconsin - Milwaukee taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/230269/math-631-university-of-wisconsin-milwaukee in Mathematics (M) at University of Wisconsin - Milwaukee.

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Date Created: 10/27/15

MATH 631632 MODERN ALGEBRA University of Wisconsin Milwaukee 20012002 draft Printed March 22 2002 Allen D Bell adbell uwmedu Chapter 1 Chapter 2 1 2 3 4 5 6 7 Chapter 3 1 2 Contents THE BASICS NUMBERS7 MATRICES7 FUNCTIONS7 RELATIONS7 AND RINGS Sets Numbers Properties of Complex Numbers Complex Calculus Commutative Rings Relations Functions Matrices Rings Formal Constructions lsomorphism and Homomorphisms Homework Exercises FACTORIZATION IN COMMUTATIVE RINGS Units7 Zero DiVisors7 and Integral Domains Factorization and Size Functions Euclidean Domains Unique Factorization Domains and Integral Elements Factorization of Polynomials Factorization in Quadratic Number Rings Homework Exercises SOME CONSTRUCTIONS FOR COMMUTATIVE RINGS Fraction Rings Congruences and Quotient Rings 73 74 76 82 90 96 108 115 125 128 iv CONTENTS 3 The Kernel and The Isomorphism Theorems 4 Ideals and Factorization 5 Homework Exercises Chapter 4 MATRICES7 LINEAR EQUATIONS7 AND DETERMINANTS 1 Matrices 2 Elementary Matrices7 Inverses7 and Linear Equations 3 Determinants 4 Inverses 5 The Characteristic and Minimal Polynomials of a Matrix 6 Homework Exercises Chapter 5 MODULES7 VECTOR SPACES7 BASES7 AND DIMENSION Modules and Vector Spaces Direct Sums Span7 Linear Independence7 and Basis Bases and Dimension 1 2 3 4 5 In nite Bases 6 Bases for Modules over Commutative Rings 7 Bases and Linear Maps 8 Homework Exercises Chapter 6 LINEAR MAPS7 MATRICES7 CHANGE OF BASIS7 AND NORMAL FORM The Matrix of a Linear Map Change of Basis Diagonalization and Eigenvectors Eigenvectors and Diagonalization Normal Form for Matrices and Operators Jordan Form 399 0 Homework Exercises Chapter 7 INNER PRODUCT SPACES 1 Inner Products 134 138 144 149 149 157 159 169 172 175 179 179 186 189 193 199 202 205 208 213 213 219 221 224 231 233 236 241 241 CONTENTS 2 Orthonormal Bases 3 Change Of Basis and Normal Form 4 Homework Exercises Chapter 8 GROUPS 1 Semigroups7 Groups7 and Subgroups Cyclic Groups and Order of Elements Homomorphisms and lsomorphisms Cosets7 Lagrange7s Theorem7 and Normal Subgroups 2 3 4 5 Quotient Groups and the lsomorphism Theorems 6 Products 7 Finite Abelian Groups 8 Homework Exercises 9 Appendix Modules and Jordan Form Chapter 9 GROUPS AND SYMMETRY 1 Geometric Symmetry 2 Abstract Symmetry and Group Actions 3 The Sylow Theorems 4 The Structure of Permutations 5 Homework Exercises Index of De nitions 244 247 254 257 257 265 268 271 277 282 285 290 298 303 303 314 322 327 331 337 These notes constitute an introduction to modern algebra They are meant to be used in tandem with my lectures They follow the same general perspective as the lectures7 but they are not identical to the lectures Sometimes I skip material in the notes other times I discuss material in class that is not in the notes7 or treat the material at greater depth in the lectures No two presentations of the same subject are identical7 even those by a single person There are a number of books covering similar material7 but none cover exactly the same ground If you wish to look at a reference book7 I recommend the book Algebra by Michael Artin If we have occasion to refer it7 we will call it Artin The rst three chapters deal mainly with the theory of commutative rings7 a gener alization of numbers Chapter 1 is a catch all reviewing and introducing material about sets7 numbers7 functions7 relations7 arithmetic mod n and polynomials It also contains the de nition of a commutative ring7 and the rst abstract tools for dealing with algebraic objects homomorphisms and isomorphisms Chapter 2 deals with the topic of factorization into irreducible elements primes in integral domains 7 we introduce the notion of a Unique Factorization Domain The theory is developed in the abstract for rings that possess a size function and a division algorithm7 and concretely for polynomial rings and rings whose elements have the form a b for integers a7b7n The former rings give an introduction to some of the tools of Algebraic Geometry7 while the latter rings introduce us to some of the tools of Algebraic Number Theory Chapter 3 deals with abstract constructions the eld of fractions of an integral domain and the quotient ring of an arbitrary commutative ring modulo an equivalence relation or an ideal In this chapter we develop some basic facts about ideals7 and introduce the notion of a Principal ldeal Domain Chapters 477 deal with linear algebra7 a eld that is central to virtually all of mathe matics Chapter 4 introduces the main computational tools7 matrices and determinants7 and develops their basic properties A formula for the inverse of a matrix is derived and the characteristic polynomial is introduced Chapter 5 introduces modules and vector spaces7 a generalization of vectors Bases are de ned and it is proven that every vector space has a basis7 and that any two bases CONTENTS 1 of a space have the same size This enables us to de ne the dimension of a vector space We also introduce linear maps and relate bases and maps In Chapter 67 we explore the relationship between matrices and linear maps This relationship is tied to the choice of a basis or bases7 and we determine the formulas for change of basis After this7 we focus on maps from a space to itself In this setting we de ne eigenvalues and eigenvectors7 and we study the possibility of diagonalizing a linear operator or a matrix We state but do not prove the theorem on the existence and uniqueness of Jordan form for operators and matrices over C In Chapter 7 we augment our algebraic de nition of vectors by introducing angle and length via an inner product This leads us to study orthonormal bases7 the analog of the perpendicular axes we are accustomed to from analytic geometry The avor of this material is more geometric and analytic than the previous chapters We prove the Spectral Theorem7 a beautiful result on diagonalization in the inner product space setting Chapters 8 and 9 begin the study of group theory In Chapter 87 we introduce the basic concepts7 such as subgroups7 cyclic groups7 homomorphism and isomorphism7 and quotient groups We give a complete description of cyclic groups7 and then we use the notion of internal direct sum to give a classi cation of nite Abelian groups We prove Lagrange7s Theorem the order of a subgroup divides the order of the whole group7 and we give some applications of it In Chapter 97 we begin by studying symmetry groups of geometric objects7 in par ticular7 polygons We give a complete list of nite symmetry groups in the plane We then turn to a more abstract kind of symmetry7 embodied by the notion of a group ac tion and permutations We see how this point of view enables us to re prove and extend some earlier results We prove the Sylow Theorems7 and begin the massive work of understanding nite groups There are problems scattered throughout the notes for the reader to ponder7 and there are more homework problems at the end of each chapter These vary a lot in dif culty7 from routine to some l7m not sure of the answer to To these you should add whatever questions occur to you while reading In addition7 not all proofs are given7 and even when proofs are given7 there are often details to be supplied ranging from trivial to dif cult Remember7 to really master mathematics you must read and learn actively ii CONTENTS These notes are not complete For example the only index is of de nitions of terms and I think I can guarantee that you will nd errors 7 of course that should only make you learn more Ideally you shouldnt believe anything I say until you7ve digested and veri ed it yourself If you have any comments or corrections 1 hope you will share them with me Gset 315 ij submatrix Aij 160 m X 71 matrix 38 150 m X 71 matrix over X 53 pgroup 322 psubgroup 322 ptorsion 286 11 31 Abelian 258 absorbency 131 action 155 315 action of X on Y 180 addition mod n 21 additive 77 adjoint 169 247 adjugate 169 af ne group 291 algebra of matrices 156 algebraic integer 93 algebraic multiplicity 230 algebraic number 93 algebraically closed 7 alternating function 162 alternating group 279 330 antire exive 29 antisymmetric 26 argument 9 associates 84 Index of De nitions 337 association 84 automorphism 63 294 basis 190 192 Bezout domain 118 bijection 32 165 bijective 32 binary operation 16 31 258 Boolean ring 67 Bounded 14 bounded above 49 bounded below 49 canonical extension 99 canonical inclusion 33 canonical inner product 243 canonical projection 129 cardinality 2 Cartesian product 3 center 264 central inversion 307 centralizer 291 318 chain 200 change of basis matrix 219 characteristic 25 characteristic polynomial 172 228 characteristic polynomial of a linear operator 228 characteristic subgroup 294 338 INDEX OF DEFINITIONS characteristic value 225 characteristic vector 225 circle group 263 circulant 175 coordinate vector 219 codomain 25 30 coef cients 190 cofactors 160 column rank 218 column space 218 column vector 37 38 150 151 common divisor 84 commutative 258 commutative diagram 215 commutative ring 17 companion matrix 301 comparabality 26 complement 20 211 complete 49 complete expansion of the determinant 166 completeness 49 complex conjugate 12 componentWise 52 composition 33 congruence 128 184 277 congruence mod a 129 congruence mod W 185 congruent 27 312 conjugacy class 318 conjugate 12 79 295 conjugate transpose 247 conjugation 316 content 102 convolution 56 coordinateWise 52 coset 132 coset determined by a 132 cosets 185 cycle 275 328 cycle structure 329 cyclic 267 cyclic subgroup generated by 265 cyclotomic polynomial 107 Dedekind domain 139 degree 56 denominator set 125 diagonal 156 diagonalizable 225 dihedral group 310 dilation 312 dimension 193 196 204 direct image 31 direct product 282 direct sum 187 282 disjoint 328 divides 6 27 83 division 18 domain 25 30 dot product 39 151 dual numbers 53 dual space 236 echelon form 158 eigenspace 228 eigenvalue 225 eigenvector 225 elementary divisors 289 elementary matrix 158 elementary row operations 157 embedding 128 empty set 2 entry 38 53 150 epimorphism 31 equivalence class 28 equivalence relation 26 equivalent 220 Euclidean domain 82 Euclidean eld of numbers 241 Euclidean motion 310 Euler function 274 evaluation at a 64 evaulation of f at a 58 even permutation 330 expansion by cofactors 161 expansion by minors 161 external direct sum 188 external laW of composition 180 factor group 278 factor module 185 factor of 27 83 factor ring 132 Fermat primes 108 Fibonacci sequence 223 eld 17 eld of fractions of 93 eld of numbers 5 nite dimensional vector spaces 194 nitely spanned 194 formal eld of fractions 126 formal fraction eld 126 formal power series 70 formal quotient eld 127 fraction eld 93 fraction ring 93 free module 202 free on 202 Frobenius homomorphism 70 function from A to B 30 INDEX OF DEFINITIONS 339 Gaussian Integers 6 general linear group 213 general linear group of degree n 259 generalized deigenspace 230 generalized Euclidean domain 90 generalized size function 90 generated 334 generated by 132 generator 267 generators and relations 259 geometric multiplicity 230 glb 49 GramSchmidt process 245 graph 25 30 greatest common divisor 84 greatest lower bound 49 group 258 group of symmetries 306 hermitian 247 250 homomorphism 61 183 269 hyperplane 307 ideal 130 idempotent 210 239 identity function 33 image 30 279 imaginary part 8 inclusion function 33 indecomposable 284 indeterminate 55 index 272 index set 3 inf 49 in mum 49 in nitesimals 53 inj ective 31 340 INDEX OF DEFINITIONS inner automorphism 294 inner product 241 inner product space 242 integral closure 95 integral domain 75 integrally closed 95 internal direct product 282 internal direct sum 188 282 intersection 3 invariant factors 289 invariant subspace 231 inverse 35 260 inverse image 31 invertible 42 155 260 involution 79 irreducible 76 81 isometry 246 304 isomorphic 61 62 183 270 isomorphism 46 61 62 183 269 isomorphism theorems 125 Jordan block 235 Jordan form 235 kernel 134 186 279 Kronecker delta 41 153 Lagrange expansion of the determinant 166 Lagrange interpolation 121 leading coef cient 56 leading term 56 least upper bound 49 left cancellable 36 left congruence modulo H 275 left coset of H determined by a 272 left domain 25 left inverse 35 260 left invertible 42 155 left module 180 length 243 328 length function 244 linear combination 190 192 linear function 162 linear map 183 linear operator 183 linear order 26 linear subset 182 linear transformation 183 linearly dependent 189 190 linearly independent 189 190 192 localization 93 lower triangular 156 lub 49 main diagonal 40 153 map 30 matrix 37 38 150 matrix of relative to B 221 matrix of the linear map 216 matrix over R 38 150 matrix units 175 maximal 138 maximal element 200 maximal With respect to the property 199 metade nition 61 method of undetermined coef cients 101 minimal polynomial 174 minors 160 mirror 306 module 180 monic 56 monoid 258 monomorphism 31 multiple 6 multiple of 27 83 multiplication mod n 21 multiplicative 77 natural extension 99 natural inclusion 33 natural projection 129 nilpotent 233 nontrivial 328 nontrivial cycle 328 nonzero divisor 74 norm 79 244 normal 247 250 normal subgroup 275 normalizer 295 322 nullity 208 218 nullspace 218 odd permutation 330 onetoone 31 onto 31 operation of X on Y 180 orbit 317 order 266 ordered basis 191 ordered pair 3 orientationpreserving 310 orientationreversing 310 orthogonal 243 247 orthogonal complement 255 orthonormal 243 pairWise relatively prime 115 partial order 26 partition 29 partitions 28 Pellls Equation 111 117 INDEX OF DEFINITIONS 341 perfect square 78 permutation 165 permutation matrix 167 perpendicular 243 PID 132 pointWise 20 52 pointWise operations 214 polar form 10 polarization 244 255 polynomial 21 polynomial equation 11 polynomial functions 64 polynomial ring 21 55 polynomials in x and y 59 power set 3 20 preorder 26 primary rational canonical form 302 prime 86 118 140 primitive 11 102 primitive nth root of unity 11 principal 132 principal ideal 132 principal ideal domain 132 principal ideal ring 132 product group 282 proper subgroup 264 proper subset 2 quaternionic group 264 quotient 47 quotient eld 93 quotient group 278 quotient group of G modulo E 278 quotient module 185 quotient ring 93 132 range 30 342 rank 204 208 218 rational canonical form 302 real part 8 reduced echelon form 159 reduction mod p 100 re ection 306 re exive 26 regular 310 regular element 74 relation between A and B 25 relation on A 25 relatively prime 11 86 remainder 21 47 right cancellable 36 right congruence 294 right congruence modulo H 272 right coset of H determined by a 272 right domain 25 right inverse 35 260 right invertible 42 155 rigid motion 304 ring 44 ring of algebraic integers 93 ring of algebraic numbers 93 ring of fractions of 93 ring of matrices 43 ring of numbers 5 ring of quaternions 45 root 58 roots of unity 10 rotational group of order n 311 row echelon form 158 row rank 218 row space 218 row vector 38 151 scalar matrix 291 INDEX OF DEFINITIONS scalar multiplication 155 180 scalars 156 semigroup With identity 258 semiring 47 sequences 4 set difference 3 set subtraction 3 sign of 166 similar 222 312 simple 325 size function 77 skewhermitian 247 250 skewsymmetric 247 skewsymmetric function 162 source 25 30 span 190 192 span V 190 special linear group of degree n 259 square 38 151 squarefree 96 108 stabilizer 317 standard regular ngon 310 strict 29 strict comparability 29 strict linear order 29 strict partial order 29 strict preorder 29 strict size function 78 strict total order 29 strictly antisymmetric 29 sub eld 18 subgroup 262 submatrix 160 submodule 181 submodule generated by 192 submodule spanned by 190 submonoid 262 subring 18 44 subset 2 subspace 181 subspace spanned by 190 192 subtraction 18 sum 187 sup 49 superset 2 support 328 supremum 49 surjective 31 Sylow subgroup 322 symmetric 26 247 symmetric bilinear form 254 symmetric difference 20 symmetric group of degree n 258 symmetric group on 71 letters 258 symmetric polynomial 331 symmetry 304 symmetry group 306 target 25 30 total degree 60 total order 26 totient 274 trace 159 transitive 26 317 translation 304 transpose 43 156 transposition 329 triangular 156 trichotomy 29 trivial linear combination 190 trivial orbit 328 trivial subgroup 264 twosided inverse 35 260 INDEX OF DEFINITIONS 343 UFD 90 unary operation 17 31 union 3 unique factorization domain 90 unit 74 unitarily diagonalizable 249 unitarily similar 248 unitary 247 unitary group of degree 1 263 unitary transformation 246 upper bound 200 upper triangular 156 value off at a 58 vector space 180 well order 46 well ordering 46 wellde ned 31 zero 58 zero divisor 74 zero ring 74

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