Notes on Abstract Algebra

314 Pages·2011·2.26 MB·English

by  John Perry

Notes on Abstract Algebra
JohnPerry
UniversityofSouthernMississippi
[e-mail]
http://www.math.usm.edu/perry/
Copyright2009JohnPerry
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 . Monoids
1. From integers to monoids...........................................................4
1. Somefactsabouttheintegers ......................................................... 5
2. Integers,monomials,andmonoids...................................................11
3. DirectProductsandIsomorphism....................................................15
Part I. Groups
2. Groups ............................................................................ 22
1. Groups...........................................................................22
2. Thesymmetriesofatriangle.........................................................29
3. Cyclicgroupsandorder ............................................................ 36
4. EllipticCurves .................................................................... 43
3. Subgroups.........................................................................47
1. Subgroups.........................................................................47
2. Cosets............................................................................51
3. Lagrange’sTheorem................................................................56
4. QuotientGroups .................................................................. 59
5. “Clockwork”groups................................................................64
6. “Solvable”groups..................................................................67
4. Isomorphisms......................................................................72
1. Fromfunctionstoisomorphisms.....................................................72
2. Consequencesofisomorphism ....................................................... 78
3. TheIsomorphismTheorem..........................................................83
4. Automorphismsandgroupsofautomorphisms.........................................87
5. Groups of permutations............................................................92
1. Permutations......................................................................92
2. Groupsofpermutations ........................................................... 101
3. Dihedralgroups .................................................................. 103
4. Cayley’sTheorem.................................................................108
5. Alternatinggroups................................................................111
i
6. The15-puzzle .................................................................... 116
6. Number theory...................................................................119
1. GCDandtheEuclideanAlgorithm.................................................119
2. TheChineseRemainderTheorem...................................................123
3. Multiplicativeclockworkgroups....................................................130
4. Euler’sTheorem..................................................................136
5. RSAEncryption..................................................................140
Part II. Rings
7. Rings.............................................................................147
1. Astructureforadditionandmultiplication..........................................147
2. IntegralDomainsandFields ....................................................... 150
3. Polynomialrings ................................................................. 155
4. Euclideandomains................................................................163
8. Ideals.............................................................................169
1. Ideals............................................................................169
2. PrincipalIdeals...................................................................175
3. Primeandmaximalideals.........................................................178
4. QuotientRings...................................................................181
5. FiniteFieldsI .................................................................... 186
6. Ringisomorphisms ............................................................... 192
7. AgeneralizedChineseRemainderTheorem..........................................197
8. Nullstellensatz....................................................................200
9. Rings and polynomial factorization................................................203
1. Thelinkbetweenfactoringandideals ............................................... 203
2. UniqueFactorizationdomains ..................................................... 206
3. FinitefieldsII ....................................................................209
4. Polynomialfactorizationinfinitefields..............................................214
5. Factoringintegerpolynomials......................................................219
10. Gröbner bases ................................................................... 224
1. Gaussianelimination............................................................225
2. Monomialorderings ............................................................. 231
3. Matrixrepresentationsofmonomialorderings ...................................... 238
4. ThestructureofaGröbnerbasis...................................................241
5. Buchberger’salgorithm...........................................................251
6. Elementaryapplications..........................................................259
11. Advanced methods of computing Gröbner bases..................................264
1. TheGebauer-Mölleralgorithm .................................................... 264
2. TheF4algorithm................................................................273
ii
3. Signature-basedalgorithmstocomputeaGröbnerbasis...............................278
Part III. Appendices
Where can I go from here?...........................................................287
Advancedgrouptheory.............................................................287
Advancedringtheory .............................................................. 287
Applications.......................................................................287
Hints to Exercises ................................................................... 288
HintstoChapter1.................................................................288
HintstoChapter2.................................................................288
HintstoChapter3.................................................................290
HintstoChapter4.................................................................291
HintstoChapter5.................................................................291
HintstoChapter6.................................................................292
HintstoChapter7.................................................................293
HintstoChapter8.................................................................294
HintstoChapter9.................................................................295
HintstoChapter10................................................................295
Index................................................................................296
References...........................................................................300
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
F
anarbitraryfield
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...
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
α
swp thesignfunctionofacycleorpermutation
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
[
Thesenotesareinspiredfromsomeofmyfavoritealgebratexts: AF05,CLO97,HA88,
]
KR00, Lau03, LP98, Rot06, Rot98 . The heritage is hopefully not too obvious, but in some
placesIfeltcompelledtocitethesource.
Thanks to the students who found typos, including (in no particular order) Jonathan
Yarber,KyleFortenberry,LisaPalchak,AshleySanders,SedrickJefferson,ShainaBarber,Blake
Watkins, and others. Special thanks go to my graduate student Miao Yu, who endured the first
draftsofChapters7,8,and10.
RogérioBritoofUniversidadedeSãoPaolomadeseveralhelpfulcomments,foundsome
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 (and therefore LATEX Lam86, Grä04 (and therefore TEX Knu84 )), along
withthepackages
◦ cc-beamer[Pip07],
◦ hyperref[RO08],
◦ AMS-LATEX[Soc02],
◦ mathdesign[Pic06],and
◦ algorithms(modifiedslightlyfromversion2006/06/02)[Bri];and
• [ ]
Inkscape Bah08 .
I’velikelyforgottensomeothernon-trivialresourcesthatIused. Letmeknowifanothercitation
belongshere.
Mywifeforeboreanumberoflatenightsattheoffice(orathome)asIworkedonthese.
AdmaioremDeigloriam.
1Inoneegregiousexample,IconnectedtoomanydotsregardingtheoriginoftheChineseRemainderTheorem.