Abstract Algebra: Basics, Polynomials, Galois Theory Categorial and Commutative Algebra [Lecture notes]

383 Pages·2005·1.309 MB·English

by  Andreas Hermann

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
[e-mail], 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 Pru¨fer. . . . . . . . . . . . . . . . . . . . . . 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¨obner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 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
clearred line). Neverthelesswetry to reachunderstandabilitybybeing 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
referenceforintermediatetoadvancedstudents,whohavealreadyhadafirst
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, ...)andnaivesettheory(Cantor’snotionofaset, 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
willleadtostandardtopicslikethefundamentaltheoremofarithmetic. 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
Pru¨fertobeprecise)wewilltreatcanonicalformtheory(e.g.Jordannormal
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¨obner bases 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,butwechosenotto. Eventhoughgrouptheoryseemstobeelementary
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.
Howeverweareaimingdirectlyfortherepresentationtheoryoffinitegroups.
Thefirstpartisconcludedbypresentingathoroughintroductiontowhat
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 -
nothingthatcanbedonewithcategorytheorycouldnothavebeenachieved
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
casearefieldextensionsincludingthebeautifultopicofGaloistheory. 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. Examples
are: linear algebra on infinite dimensional vector-spaces,representation the-
ory of infinite groups and Galois theory of infinite field extensions. Another
naturalextensionwouldbetointroducetheZariskitoplogyonthespectrum
of a ring, which would lead to the theory of schemes directly. The scope
of this text is purely algebraic and hence we will stop at the point where
toplogoy sets in (and give hints for further readings only).
The Two Parts
Mathematics has a peculiarity to it: there are problems (and answers) that
are easy to understand but hard to prove. The most famous example is
Fermat’s Last Theorem - the statement (for any n ≥ 3 there are no non-
trivial integers (a,b,c) ∈ Z3 that satisfy the equation an + bn = cn) can
be understood by anyone. Yet the proof is extremely hard to provide. Of
course this theorem has no value of its own (it is the proof that contains
deep insights into the structure of mathematics), but this is no general rule.
E.g.thetheoremofWedderburn(everyfiniteskew-fieldisafield)iseasyand
useful, but its proof will be perfomed using a beautiful trick-computation
(requiring the more advanced method of cyclotomic polynomials).
Thus we have chosen an uncostomary approach: we have seperated the
truth (i.e. definitions, examples and theorems) from their proofs. This en-
ables us to present the truth in a stringent way, that allows the reader to
get a feel for the mathematical objects displayed. Most of the proofs could
7
have been given right away, but in several cases the proof of a statement can
only be done after we have developped the theory further. Thus the sequel
of theorems may (and will) be different from the order in which they are
proved. Hence the two parts.
Our Best Advice
It is a well-known fact, that some proofs are just computational and only
contain little (or even no) insight into the structure of mathematics. Others
are brilliant, outstanding insights that are of no lesser importance than the
theorem itself. Thus we have already included remarks of how the proof
works in the first part of this book. And our best advice is to read a section
entirely to get a feel for the objects involved - only then have a look at the
proofs that have been recommended. Ignore the other proofs, unless you
have to know about them, for some reason. At several occasions this text
contains the symbols (♦) and ((cid:165)) . These are meant to guide the reader in
the following way:
(♦) As we have assorted the topics covered thematically (paying little at-
tentiontothesequelofproofs)itmighthappenthatacertainexample
or theorem is beyond the scope of the theory presented so far. In this
case the reader is asked to read over it lightly (or even skip it entirely)
and come back to it later (when he has gained some more experience).
((cid:165)) On some very rare occasions we will append a theorem without giving
aproof(iftheproofisbeyondthescopeofthistext). Suchaninstance
will be marked by the black box symbol. In this case we will always
give a complete reference of the most readable proof the author is
aware of. And this symbol will be hereditarily, that is once we use
a theorem that has not been proved any other proof relying on the
unproved statement will also be branded by the black box symbol.
8
0.2 Notation and Symbols
Conventions
We now wish to include a set of the frequently used symbols, conventions
and notations. In particular we clarify the several domains of numbers.
• First of all we employ the nice convention (introduced by Halmos) to
write iff as an abbreviation for if and only if.
• We denote the set of natural numbers - i.e. the positive integers in-
cluding zero - by N := {0,1,2,3,...}. Further for any two integers
a,b ∈ Z we denote the interval of integer numbers ranging from a to
b by a...b := {k ∈ Z | a ≤ k ≤ b}.
• We will denote the set of integers by Z = N∪(−N), and the rationals
by Q = {a/b | a,b ∈ Z,b 6= 0}. Whereas Z will be taken for granted,
Q will be introduced as quotient field of Z.
• The reals will be denoted by R and we will present an example of how
they can be defined (without proving their properties however). The
complex numbers will be denoted by C = {a+ib | a,b ∈ R} and we
will present several ways of constructing them.
• (♦) We will sometimes use the Kronecker-Symbol δ(a,b) (in the liter-
ature this is also denoted by δ ), which is defined to be
a,b
(cid:189)
1 if a = b
δ(a,b) = δ :=
a,b 0 if a 6= b
Inmostcasesaandb ∈ Zwillbeintegersand0,1∈ Zwillbeintegers,
too. Butingeneralwearegivensomering(R,+,·)anda,b ∈ R. Then
the elements 0 and 1 ∈ R on the right hand side are taken to be the
zero-element 0 and unit-element 1 of R again.
• We will write A ⊆ X to indicate that A is a subset of X and A ⊂ X
will denote strict inclusion (i.e. A ⊆ X and there is some x ∈ X with
x 6∈ A). For any set X we denote its power set (i.e. the set of all its
subsets) by P(X) := {A | A ⊆ X}. And for a subset A ⊆ X we
denote the complement of A in X by CA := X \A.
• Listing several elements x ,...,x ∈ X of some set X, we do not
1 n
requirethesex tobepairwisedistict(e.g.x = x mightwellhappen).
i 1 2
Yetifweonlygiveexplicitnamesx totheelementsofsomepreviously
i
given subset A = {x ,...,x } ⊆ X we already consider the x to be
1 n i
pairwise distinct (that is x = x implies i = j). Note that if the
i j
x (not the set {x ,...,x }) is given then {x ,...,x } may hence
i 1 n 1 n
contain fewer than n elements!
9
(cid:83)
• Given an arbitary set of sets A one defines the grand union A and
(cid:84)
the grand intersection A to be the set consisting of all elements a
that are contained in one (resp. all) of the sets A ∈ A, formally
(cid:91)
A := {a | ∃A ∈ A : a ∈ A}
(cid:92)
A := {a | ∀A ∈ A : a ∈ A}
(cid:84)
Notethat Aonlyisawell-definedset,ifA 6= ∅isnon-empty. Awell-
knownspecialcaseofthisisthefollowing: consideranytwosetsAand
(cid:83) (cid:84)
B and let A := {A,B}. Then A∪B = A and A∩B = A. This
notion is just a generalisation of the ordinary union and intersection
to arbitary collections of sets A.
• If X and Y are any sets then we will denote the set of all functions
from X to Y by F(X,Y) = YX = {f | f : X → Y }. And for any
such function f : X → Y : x → f(x) we will denote its graph by
(note that from the set-theoretical point of view f is its graph)
Γ(f) := {(x,f(x)) | x ∈ X} ⊆ X ×Y
• Once we have defined functions, it is easy to define arbitarycarthesian
products. That is let I 6= ∅ be any non-empty set and for any i ∈ I let
X be another set. Let us denote the union of all the X by X, that is
i i
(cid:91)
X := X = {x | ∃i ∈ I : x ∈ X }
i i
i∈I
Then the carthesian product of the X consisits of all the functions
i
x : I → X such that for any i ∈ I we have x := x(i) ∈ X . Note that
i i
thereby it is customary to write (x ) in place of x. Formally
i
(cid:89)
X := {x : I → X : i 7→ x | ∀i ∈ I : x ∈ X }
i i i i
i∈I
• Let X 6= ∅ be a nonempty set, then a subset of the form R ⊆ X ×X
said to be a relation on X. And in this case it is customary to write
xRy instead of (x,y) ∈ R. This notation will be used primarily for
partial orders and equivalence relations (see below).
• Consider any nonempty set X 6= ∅ again. Then a relation ∼ on X is
saidtobeanequivalence relationonX, iffitisreflexive, symmetric
and transitive. Formally that is for any x, y and z ∈ X we get
x = y =⇒ x ∼ y
x ∼ y =⇒ y ∼ x
x ∼ y, y ∼ z =⇒ x ∼ z
10