Abstract Algebra Basics, Polynomials, Galois Theory Categorial and Commutative Algebra by Andreas Hermann June 22, 2005 (383 pages) If you have read this text I would like to invite you to contribute to it. Comments, corrections and suggestions are very much appreciated, at [email protected], or visit my homepage at www.mathematik.uni-tuebingen.de/ab/algebra/index.html This book is dedicated to the entire mathematical society. To all those who contribute to mathematics and keep it alive by teaching it. Contents 0.1 About this Book . 5 0.2 Notation and Symbols . 9 0.3 Mathematicians at a Glance . 14 1 Groups and Rings 17 1.1 Defining Groups . 17 1.2 Defining Rings . 22 1.3 Examples . 26 1.4 First Concepts . 35 1.5 Ideals . 41 1.6 Homomorphisms . 54 1.7 Products . 61 2 Commutative Rings 67 2.1 Maximal Ideals . 67 2.2 Prime Ideals . 73 2.3 Radical Ideals . 77 2.4 Noetherian Rings . 82 2.5 Unique Factorisation Domains . 88 2.6 Principal Ideal Domains . 100 2.7 Lasker-Noether Decomposition . 111 2.8 Finite Rings . 119 2.9 Localisation . 122 2.10 Local Rings . 134 2.11 Dedekind Domains . 140 3 Modules 146 3.1 Defining Modules . 146 3.2 First Concepts . 158 3.3 Free Modules . 167 3.4 Homomorphisms . 177 3.5 Rank of Modules . 178 3.6 Length of Modules . 179 3.7 Localisation of Modules . 180 2 4 Linear Algebra 181 4.1 Matices . 181 4.2 Elementary Matrices . 182 4.3 Linear Equations . 183 4.4 Determinants . 184 4.5 Rank of Matrices . 185 4.6 Canonical Forms . 186 5 Structure Theorems 187 5.1 Asociated Primes . 188 5.2 Primary Decomposition . 189 5.3 The Theorem of Pr¨ufer. 190 6 Polynomial Rings 191 6.1 Monomial Orders . 191 6.2 Graded Algebras . 197 6.3 Defining Polynomials . 202 6.4 The Standard Cases . 203 6.5 Roots of Polynomials . 204 6.6 Derivation of Polynomials . 205 6.7 Algebraic Properties . 206 6.8 Gr¨obnerBases . 207 6.9 Algorithms . 208 7 Polynomials in One Variable 209 7.1 Interpolation . 210 7.2 Irreducibility Tests . 211 7.3 Symmetric Polynomials . 212 7.4 The Resultant . 213 7.5 The Discriminant . 214 7.6 Polynomials of Low Degree . 215 7.7 Polynomials of High Degree . 216 7.8 Fundamental Theorem of Algebra . 217 8 Group Theory 218 9 Multilinear Algebra 219 9.1 Multilinear Maps . 219 9.2 Duality Theory . 220 9.3 Tensor Product of Modules . 221 9.4 Tensor Product of Algebras . 222 9.5 Tensor Product of Maps . 223 9.6 Differentials . 224 3 10 Categorial Algebra 225 10.1 Sets and Classes . 225 10.2 Categories and Functors . 226 10.3 Examples . 227 10.4 Products . 228 10.5 Abelian Categories . 229 10.6 Homology . 230 11 Ring Extensions 231 12 Galois Theory 232 13 Graded Rings 233 14 Valuations 234 15 Proofs - Fundamentals 236 16 Proofs - Rings and Modules 246 17 Proofs - Commutative Algebra 300 4 0.1 About this Book The Aim of this Book Mathematics knows two directions - analysis and algebra - and any math- ematical discipline can be weighted how analytical (resp. algebraical) it is. Analysis is characterized by having a notion of convergence that allows to approximate solutions (and reach them in the limit). Algebra is character- ized by having no convergence and hence allowing finite computations only. This book now is meant to be a thorough introduction into algebra. Likewise every textbook on mathematics is drawn between two pairs of extremes: (easy understandability versus great generality) and (complete- ness versus having a clear line of thought). Among these contrary poles we usually chose (generality over understandability) and (completeness over a clear red line). Nevertheless we try to reach understandability by being very precise and accurate and including many remarks and examples. At last some personal philosophy: a perfect proof is like a flawless gem - unbreakably hard, spotlessly clear, flawlessly cut and beautifully displayed. In this book we are trying to collect such gemstones. And we are proud to claim, that we are honest about where a proof is due and present complete proofs of almost every claim contained herein (which makes this textbook very different from most others). This Book is Written for many differen kinds of mathematicans: primarily is meant to be a source of reference for intermediate to advanced students, who have already had a first contact with algebra and now closely examine some topic for their seminars, lectures or own thesis. But because of its great generality and completeness it is also suited as an encyclopedia for professors who prepare their lectures and researchers who need to estimate how far a certain method carries. Frankly this book is not perfectly suited to be a monograph for novices to mathematics. So if you are one we think you can greatly profit from this book, but you will propably have to consult additional monographs (at a more introductory level) to help you understand this text. Prerequisites We take for granted, that the reader is familiar with the basic notions of naive logic (statements, implication, proof by contradiction, usage of quan- tifiers, ... ) and naive set theory (Cantor’s notion of a set, functions, parially ordered sets, equivalence relations, Zorn’s Lemma, ... ). We will present a short introduction to classes and the NBG axioms when it comes to cate- gory theory. Further we require some basic knowledge of integers (including proofs by induction) and how to express them in decimal numbers. We will sometimes use the field of real numbers, as they are most propably well- 5 known to the reader, but they are not required to understand this text. Aside from these prerequesites we will start from scratch. Topics Covered We start by introducing groups and rings, immediatly specializing on rings. Of general ring theory we will introduce the basic notions only, e.g. the isomorphism theorems. Then we will turn our attention to commutative rings, which will be the first major topic of this book: we closely study maximal ideals, prime ideals, intersections of such (radical ideals) and the relations to localisation. Further we will study rings with chain conditions (noetherian and artinian rings) including the Lasker-Noether theorem. This will lead to standard topics like the fundamental theorem of arithmetic. And we conclude commutative ring theory by studying discrete valuation rings and Dedekind domains. Then we will turn our attention to modules, including rank, dimension and length. We will see that modules are a natural and powerful generalisa- tion of ideals and large parts of ring theory generalize to this setting, e.g. lo- calisation and primary decomposition. Module theory naturally leads to linear algebra, i.e. the theory of matrix representations of a homomorphism of modules. Applying the structure theorems of modules (the theorem of Pr¨uferto be precise) we will treat canonical form theory (e.g. Jordan normal form). Next we will study polynomials from top down: that is we introduce general polynomial rings (also known as group algebras) and graded alge- bras. Only then we will regard the more classical problems of polynomials in one variable and their solvability. Finally we will regard polynomials in sevreral variables again. Using Gr¨obnerbases it is possible to solve abstract algebraic questions by purely computational means. Then we will return to group theory: most textbooks begin with this topic, but we chose not to. Even though group theory seems to be elementary and fundamental this is notquite true. In fact it heavily relies on arguments like divisibility and prime decomposition in the integers, topics that are native to ring theory. And commutative groups are best treated from the point of view of module theory. Never the less you might as well skip the previous sections and start with group theory right away. We will present the standard topics: the isomorphism theorems, group actions (including the formula of Burnside), the theorems of Sylow and lastly the p-q-theorem. However we are aiming directly for the representation theory of finite groups. The first part is concluded by presenting a thorough introduction to what is called multi-linear algebra. We will study dual pairings, tensor products of modules (and algebras) over a commutative base ring, derivations and the module of differentials. 6 Thus we have gathered a whole buch of seperate theories - and it is time for a second structurisation (the first structurisation being algebra itself). We will introduce categories, functors, equivalency of categories, (co-)products and so on. Categories are merely a manner of speaking - nothing that can be done with category theory could not have been achieved without. Yet the language of categories presents a unifying concept for all the different branches of mathematics, extending far beyond algebra. So we will first recollect which part of the theory we have established is what in the categorial sense. And further we will present the basics of abelian categories as a unifying concept of all those seperate theories. We will then aim for some more specialized topics: At first we will study ring extensions and the dimension theory of commutative rings. A special case are field extensions including the beautiful topic of Galois theory. After- wards we turn our attention to filtrations, completions, zeta-functions and the Hilbert-Samuel polynomial. Finally we will venture deeper into number theory: studying valuations theory up to the theorem of Riemann-Roche for number fields. Topics not Covered There are many instances where, dropping a finiteness condition, one has to introduce some topology in order to pursue the theory further.