Algebra: monomials and polynomials [lecture notes]

324 Pages·2012·1.556 MB·English

by  Perry J.

Algebra: Monomials and Polynomials
JohnPerry
UniversityofSouthernMississippi
[email protected]
http://www.math.usm.edu/perry/
Copyright2012JohnPerry
www.math.usm.edu/perry/
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Table of Contents
Reference sheet for notation ........................................................... iv
A few acknowledgements .............................................................. vi
Preface................................................................................vii
Overview ........................................................................... vii
Three interesting problems.............................................................1
Part I. Monoids
1. From integers to monoids............................................................4
1. Somefactsabouttheintegers..........................................................6
2. Integers,monomials,andmonoids....................................................11
3. DirectProductsandIsomorphism .................................................... 16
Part II. Groups
2. Groups.............................................................................24
1. Groups............................................................................24
2. Thesymmetriesofatriangle.........................................................32
3. Cyclicgroupsandorder.............................................................38
4. Therootsofunity...................................................................45
5. EllipticCurves.....................................................................48
3. Subgroups..........................................................................52
1. Subgroups ......................................................................... 52
2. Cosets.............................................................................56
3. Lagrange’sTheorem ................................................................ 61
4. QuotientGroups...................................................................64
5. “Clockwork”groups ................................................................ 70
6. “Solvable”groups...................................................................73
4. Isomorphisms ...................................................................... 79
1. Homomorphisms...................................................................79
2. Consequencesofisomorphism........................................................85
3. TheIsomorphismTheorem .......................................................... 90
4. Automorphismsandgroupsofautomorphisms.........................................94
5. Groups of permutations ............................................................ 98
1. Permutations ...................................................................... 98
2. Cyclenotation....................................................................101
3. Dihedralgroups...................................................................109
4. Cayley’sTheorem ................................................................. 114
5. Alternatinggroups ................................................................ 117
6. The15-puzzle.....................................................................121
i
6. Number theory....................................................................125
1. TheEuclideanAlgorithm .......................................................... 125
2. TheChineseRemainderTheorem ................................................... 129
3. TheFundamentalTheoremofArithmetic............................................136
4. Multiplicativeclockworkgroups.....................................................138
5. Euler’sTheorem...................................................................143
6. RSAEncryption .................................................................. 146
Part III. Rings
7. Rings..............................................................................154
1. Astructureforadditionandmultiplication .......................................... 154
2. IntegralDomainsandFields........................................................158
3. Polynomialrings..................................................................163
4. Euclideandomains................................................................170
8. Ideals..............................................................................177
1. Ideals............................................................................177
2. PrincipalIdealDomains...........................................................183
3. Primeandmaximalideals..........................................................187
4. QuotientRings ................................................................... 190
5. FiniteFieldsI.....................................................................195
6. Ringisomorphisms................................................................201
9. Factorization......................................................................208
1. Thelinkbetweenfactoringandideals................................................208
2. UniqueFactorizationdomains......................................................211
3. FinitefieldsII.....................................................................213
4. Extendingaringbyaroot..........................................................220
5. Polynomialfactorizationinfinitefields .............................................. 223
6. Factoringintegerpolynomials.......................................................231
10. Roots of multivariate polyomials..................................................234
1. Gaussianelimination.............................................................235
2. Monomialorderings..............................................................241
3. Matrixrepresentationsofmonomialorderings.......................................248
4. ThestructureofaGröbnerbasis....................................................251
5. Buchberger’salgorithm............................................................261
6. Nullstellensatz...................................................................270
7. Elementaryapplications .......................................................... 272
11. Advanced methods of computing Gröbner bases .................................. 277
1. TheGebauer-Mölleralgorithm.....................................................277
2. TheF4algorithm.................................................................286
3. Signature-basedalgorithmstocomputeaGröbnerbasis...............................291
Part III. Appendices
ii
Where can I go from here? ........................................................... 300
Advancedgrouptheory..............................................................300
Advancedringtheory...............................................................300
Applications ....................................................................... 300
Hints to Exercises....................................................................301
HintstoChapter1..................................................................301
HintstoChapter2..................................................................302
HintstoChapter3..................................................................303
HintstoChapter4..................................................................304
HintstoChapter5..................................................................305
HintstoChapter6..................................................................305
HintstoChapter7..................................................................306
HintstoChapter8..................................................................307
HintstoChapter9..................................................................308
HintstoChapter10.................................................................308
Index................................................................................309
References ........................................................................... 312
iii
Reference sheet for notation
[r] theelement r +nZofZ
n
〈 〉
g thegroup(orideal)generatedby g
A thealternatinggrouponthreeelements
3
(cid:47)
A G forG agroup,AisanormalsubgroupofG
(cid:47)
A R for Raring,Aisanidealof R (cid:112)
C thecomplexnumbers{a+bi : a,b ∈Cand i = −1}
[G,G] commutatorsubgroupofagroupG
[x,y] for x and y inagroupG,thecommutatorof x and y
Conj (H) thegroupofconjugationsof H bya
a
conj (x) theautomorphismofconjugationby g
g
D thesymmetriesofatriangle
3
|
d n d divides n
degf thedegreeofthepolynomial f
D thedihedralgroupofsymmetriesofaregularpolygonwith n sides
n
D (R) thesetofalldiagonalmatriceswhosevaluesalongthediagonalisconstant
n
Z
d thesetofintegermultiplesof d
f (G) for f ahomomorphismandG agroup(orring),theimageofG
Frac(R) thesetoffractionsofacommutativering R
G/A thesetofleftcosetsofA
\
G A thesetofrightcosetsofA
gA theleftcosetofAwith g
∼
G =H G isisomorphicto H
GL (R) thegenerallineargroupofinvertiblematrices
m
(cid:81)n G theordered n-tuplesofG ,G ,...,G
i=1 i 1 2 n
×
G H theorderedpairsofelementsofG and H
gz forG agroupand g,z ∈G,theconjugationof g by z,or zgz−1
<
H G forG agroup, H isasubgroupofG
kerf thekernelofthehomomorphism f
lcm(t,u) theleastcommonmultipleofthemonomials t and u
lm(p) theleadingmonomialofthepolynomial p
lv(p) theleadingvariableofalinearpolynomial p
M
thesetofmonomialsinonevariable
M
thesetofmonomialsin n variables
n
N+
thepositiveintegers
N (H) thenormalizerofasubgroup H ofG
G
N { }
thenaturalorcountingnumbers 0,1,2,3...
Ω the nthrootsofunity;thatis,allrootsofthepolynomial xn−1
n
ord(x) theorderof x
P thepointatinfinityonanellipticcurve
∞
Q thegroupofquaternions
8
Q therationalnumbers{a : a,b ∈Zand b (cid:54)=0}
b
R/A for R a ring and Aan ideal subring of R, R/Ais the quotient ring of R with
respecttoA
〈 〉
r ,r ,...,r theidealgeneratedby r ,r ,...,r
1 2 m 1 2 m
R
therealnumbers,thosethatmeasureanylengthalongaline
Rm×m m×m matriceswithrealcoefficients
R[x] polynomialsinonevariablewithrealcoefficients
R[x ,x ,...,x ] polynomialsin n variableswithrealcoefficients
1 2 n
R[x ,x ,...,x ] theringofpolynomialswhosecoefficientsareinthegroundring R
1 2 n
S thegroupofallpermutationsofalistof n elements
n
×
S T theCartesianproductofthesets S andT
tts(p) thetrailingtermsof p
Z(G) centralizerofagroupG
Z∗ Z
thesetofelementsof thatarenot zerodivisors
n n
Z Z Z Z
/n quotientgroup(resp. ring)of modulothesubgroup(resp. ideal) n
Z { − }
(cid:112) theintegers ..., 1,0,1,2,(cid:112)...
(cid:148) (cid:151)
Z − −
5 theringofintegers,adjoin 5
Z Z Z
thequotientgroup /n
n
v
A few acknowledgements
[
These notes are inspired from some of my favorite algebra texts: AF05, CLO97, HA88,
]
KR00,Lau03,LP98,Rot06,Rot98 . Theheritageishopefullynottooobvious,butinsomeplaces
Ifeltcompelledtocitethesource.
Thanks to the students who found typos, including (in no particular order) Jonathan
Yarber, Kyle Fortenberry, Lisa Palchak, Ashley Sanders, Sedrick Jefferson, Shaina Barber, Blake
Watkins,KrisKatterjohn,TaylorKilman,EricGustaffson,PatrickLambert,andothers. Special
thanksgotomygraduatestudentMiaoYu,whoenduredthefirstdraftsofChapters7,8,and10.
Rogério Brito of Universidade de São Paolo made several helpful comments, found some
nastyerrors1,andsuggestedsomeoftheexercises.
Ihavebeenluckytohavehadgreatalgebraprofessors;inchronologicalorder:
•
VanessaJobatMarymountUniversity;
•
AdrianRiskinatNorthernArizonaUniversity;
•
andatNorthCarolinaStateUniversity:
◦
KwangilKoh,
◦
HoonHong,
◦
ErichKaltofen,
◦
MichaelSinger,and
◦
AgnesSzanto.
Boneheaded innovations of mine that looked good at the time but turned out bad in practice
shouldnotbeblamedonanyoftheindividualsorsourcesnamedabove. Afterall,theyevaluated
previousworkofmine,sotheconceptthatImightsaysomethingdumbwon’tcomeasasurprise
tothem,andtheytriedveryhardtocuremeofthathabit. Thisisnotapeer-reviewedtext, which
iswhyyouhaveasupplementarytextinthebookstore.
Thefollowingsoftwarehelpedpreparethesenotes:
• [ ]
Sage3.xandlater Ste08 ;
• [ ] [ ] [ ]
Lyx Lyx (andthereforeLATEX Lam86,Grä04 (andthereforeTEX Knu84 )),alongwith
thepackages
◦ cc-beamer[Pip07],
◦ hyperref[RO08],
◦ AMS-LATEX[Soc02],
◦ mathdesign[Pic06],
◦ thmtools,and
◦ algorithms(modifiedslightlyfromtheversionreleased2006/06/02)[Bri];and
• [ ]
Inkscape Bah08 .
I’velikelyforgottensomeothernon-trivialresourcesthatIused. Letmeknowifanothercitation
belongshere.
Mywifeforeboreanumberoflatenightsattheoffice(orathome)asIworkedonthese.
AdmaioremDeigloriam.
1Inoneegregiousexample,IconnectedtoomanydotsregardingtheoriginoftheChineseRemainderTheorem.
Preface
This is not a textbook.
Okay,youask,whatisit,then?
ThesearenotesIusewhenteachingclass.
Butitlookslikeatextbook.
Fine. Sosueme. —no,wait. Letmetrytoexplain. Atwo-semestersequenceonmodern
algebraoughttointroducestudentstothefundamentalaspectsofgroupsandrings. That’salready
abitemorethanmostcanchew,andIhavedifficultycoveringeventhestuffIthinkisnecessary.
Unfortunately, most every algebra text I’ve encountered expend far too much effort in the first
50–100pageswithmaterialthatisnotalgebra. Theusualculpritisnumbertheory,butitisbyno
meansthesoleoffender. Whohasthatkindoftime?
Thenthere’s thewhole argumentabout whetherto startwith groups, rings, semigroups,
ormonoids. Desiringamixofsimplicityandutility,Idecidedtowriteoutsomenotesthatwould
getmeintogroupsassoonaspossible. Voilà.
Youstillhaven’texplainedwhyitlookslikeatextbook.
That’s because I wanted to organize, edit, rearrange, modify, and extend my notes easily.
Ialsowantedthemindigitalform,sothat(a)Icouldreadthem,2 and(b)I’dbelesslikelytolose
them. IusedasoftwareprogramcalledLyx,whichbuildsonLATEX;seetheAcknowledgments.
WhatifI’dpreferanactualtextbook?
Seethesyllabus.
Overview
These notes have two major parts: in one, we focus on an algebraic structure called a
group; in the other, we focus on a special kind of group, a ring. In the first semester, therefore,
wewanttocoverChapters2–5. Sincearigorousapproachrequiressomesortofintroduction,we
reviewsomebasicsoftheintegersandthenaturalnumbers–butonlytosolidifythefoundation
ofwhatstudentshavealreadylearned;wedonotdelveintonumbertheoryproper.
Some of these ideas fit well with monomials, which I study “on occasion”. In algebra, a
boringdiscussionofintegersandmonomialsnaturallyturnsintoafascinatingstoryofthebasics
ofmonoids,whichactuallymakesforagentleintroductiontogroups. Iyieldedtotemptationand
threwthatin. Thatshouldmakeslifeeasierlateron,anyway;abrief glanceatmonoids,focusing
on commutative monoids without requiring commutativity, allows us to introduce prized ideas
that will be developed in much more depth with groups, only in a context with which students
are far more familiar. Repetitio mater studiorum, and all that.3 We restrict ourselves to the easier
notions,sincethepointistogettogroups,andquickly.
Ideally, we’d also cover Chapter 6, but one of the inexorable laws of life is that the older
onegets, thefastertimepasses. Tempusfugit, andallthat.4 Thecorollaryforaprofessoristhata
2Youthinkyouhaveproblemsreadingmyhandwriting?Ihavetolivewithit!
3University students would do well to remember another Latin proverb, Vinum memoriæmors. I won’t provide
a translation here; look it up, you chumps! Once upon a time, a proper education included familiarity with the
wisdomofourancestors;thesedays,youcangetawaywithGooglingit.Oddly,recentstudiessuggestthattheLatin
phraseshouldbeupdated:Googlememoriæmors.
4Latinisnotaprerequisiteofthiscourse,butitshouldbe.Googleit!
semestergrowsshorterwitheachpassingyear,whichimpliesthatwecoverlesseveryyear. Ihave
noideawhy.
Inthesecondsemester,wedefinitelycoverChapters6through8,alongwithatleastoneof
Chapter9or10. Chapter11isnotapartofeithercourse,butIincludeditforstudents(graduate
or undergraduate) who want to pursue a research project, and need an introduction that builds
on what came before. As of this writing, some of those chapters still need major debugging, so
don’ttakeanythingyoureadtheretooseriously.
Not much of the material can be omitted. Within each chapter, many examples are used
and reused; this applies to exercises, as well. Textbooks often avoid this, in order to give instruc-
torsmoreflexibility;Idon’tcareaboutotherinstructors’pointsofview,soIdon’tmindputting
into the exercises problems that I return to later in the notes. We try to concentrate on a few
important examples, re-examining them in the light of each new topic. One consequence is that
ringscannotbetaughtindependentlyfromgroupsusingthesenotes.
Togiveyouaheads-up,thefollowingmaterialwillprobablybeomitted.
•
I really like the idea of placing elliptic curves (Section 2.5) early in the class. Previous edi-
tionsofthesenoteshadtheminthesectionimmediatelyaftertheintroductionofgroups! It
gives students an immediate insight into how powerful abstraction can be. Unfortunately,
Ihaven’tyetbeenabletogetgoingfastenoughtogetthemdone.
•
Groupsofautomorphisms(Section4.4)aregenerallyconsideredoptional.
•
Ihavenotinthepasttaughtsolvablegroups(Section3.6),buthopetodosoeventually.
•
Ihavesometimesnotmadeitpastalternatinggroups(Section5.5). ConsideringthatIused
to be able to make it to the RSA algorithm (Section 6.6), that does not mean we won’t get
there, especially since I’ve simplified the beginning. That was before I added the stuff on
monoids,though...
•
The discussion of the 15-puzzle is simplified from other places I’ve found it, nonstandard,
anddefinitelyoptional.
viii