Foundations of Nonlinear Algebra PDF
Preview Foundations of Nonlinear Algebra
Foundations of Nonlinear Algebra JohnPerry UniversityofSouthernMississippi [email protected] http://www.math.usm.edu/perry/ Copyright2012JohnPerry www.math.usm.edu/perry/ CreativeCommonsAttribution-Noncommercial-ShareAlike3.0UnitedStates Youarefree: • toShare—tocopy,distributeandtransmitthework • toRemix—toadaptthework Underthefollowingconditions: • Attribution—Youmustattributetheworkinthemannerspecifiedbytheauthororlicensor (butnotinanywaythatsuggeststhattheyendorseyouoryouruseofthework). • Noncommercial—Youmaynotusethisworkforcommercialpurposes. • Share Alike—If you alter, transform, or build upon this work, you may distribute the re- sultingworkonlyunderthesameorsimilarlicensetothisone. 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Thebestwaytodothisiswithalinktothiswebpage: http://creativecommons.org/licenses/by-nc-sa/3.0/ us/legalcode Table of Contents Reference sheet for notation ........................................................... vi A few acknowledgements.............................................................viii Preface................................................................................ix Overview............................................................................ix Tothestudent........................................................................x Howtosucceedatalgebra Waysthesenotestrytohelpyousucceed Some interesting problems ............................................................. 1 Nimfinity Acardtrick Internetcommerce Factorization Conclusion 0. Foundations.........................................................................4 1. Setsandrelations ................................................................... 4 Sets Relations Binaryoperations Orderings 2. Division .......................................................................... 13 TheDivisionTheorem Equivalenceclasses 3. Linearalgebra.....................................................................20 Matrices Lineartransformations Determinants Proofsofsomepropertiesofdeterminants..............................................30 Part I. Monoids and groups 1. Monoids............................................................................35 1. Fromintegersandmonomialstomonoids...........................................35 Monomials M N Similaritiesbetween and Monoids 2. Isomorphism......................................................................44 3. Directproducts....................................................................47 4. AbsorptionandtheAscendingChainCondition.....................................51 Absorption Dickson’sLemmaandtheAscendingChainCondition AlookbackattheHilbert-Dicksongame i 2. Groups.............................................................................58 1. Groups ........................................................................... 58 Precisedefinition,firstexamples Orderofagroup,Cayleytables Otherelementarypropertiesofgroups 2. Thesymmetriesofatriangle........................................................66 Intuitivedevelopmentof D 3 Detailedproofthat D containsallsymmetriesofthetriangle 3 3. Cyclicgroupsandorderofelements.................................................76 Cyclicgroupsandgenerators Theorderofanelement 4. Therootsofunity.................................................................83 Imaginaryandcomplexnumbers Thecomplexplane Rootsofunity 3. Subgroups..........................................................................94 1. Subgroups ........................................................................ 94 2. Cosets............................................................................99 Theidea PropertiesofCosets 3. Lagrange’sTheorem..............................................................103 4. QuotientGroups.................................................................107 “Normal”subgroups Quotientgroups 5. “Clockwork”groups..............................................................115 4. Isomorphisms ..................................................................... 121 1. Homomorphisms ................................................................ 121 Groupisomorphisms Propertiesofgrouphomorphism 2. Consequencesofisomorphism.....................................................128 Isomorphismisanequivalencerelation Isomorphismpreservesbasicpropertiesofgroups Isomorphismpreservesthestructureofsubgroups 3. TheIsomorphismTheorem.......................................................133 Motivatingexample TheIsomorphismTheorem 4. Automorphismsandgroupsofautomorphisms ..................................... 138 Theautomorphismgroup 5. Groups of permutations...........................................................143 1. Permutations.....................................................................143 Permutationsasfunctions Groupsofpermutations 2. Cyclenotation ................................................................... 147 ii Cycles Cyclearithmetic Permutationsascycles 3. Dihedralgroups..................................................................156 Fromsymmetriestopermutations D and S n n 4. Cayley’sTheorem................................................................161 5. Alternatinggroups ............................................................... 165 Transpositions Evenandoddpermutations Thealternatinggroup 6. The15-puzzle....................................................................170 6. Number theory....................................................................173 1. TheGreatestCommonDivisor....................................................173 Commondivisors TheEuclideanAlgorithm Bezout’sidentity 2. TheChineseRemainderTheorem ................................................. 179 ThesimpleChineseRemainderTheorem AgeneralizedChineseRemainderTheorem 3. TheFundamentalTheoremofArithmetic..........................................185 4. Multiplicativeclockworkgroups...................................................188 Z Multiplicationin n Zerodivisors Z∗ Meet n 5. Euler’sTheorem..................................................................193 Euler’sTheorem Computingϕ(n) Fastexponentiation 6. RSAEncryption ................................................................. 197 Descriptionandexample Theory Part II. Rings 7. Rings..............................................................................206 1. Astructureforadditionandmultiplication.........................................206 2. IntegralDomainsandFields.......................................................211 Twoconvenientkindsofrings Thefieldoffractions 3. Polynomialrings ................................................................. 216 Fundamentalnotions Propertiesofpolynomials 4. Euclideandomains................................................................225 Divisionofpolynomials iii Euclideandomains 8. Ideals..............................................................................231 1. Ideals............................................................................231 Definitionandexamples Propertiesandelementarytheory 2. PrincipalIdealDomains...........................................................238 Principalidealdomains NoetherianringsandtheAscendingChainCondition 3. CosetsandQuotientRings........................................................243 Thenecessityofideals Usinganidealtocreateanewring 4. Whenisaquotientringanintegraldomainorafield?................................248 Maximalandprimeideals Acriterionthatdetermineswhenaquotientringisanintegraldomainorafield 5. Ringisomorphisms...............................................................253 Ringhomomorphismsandtheirproperties Theisomorphismtheoremforrings Aconstructionofthecomplexnumbers Part III. Applications 9. Roots of univariate polynomials ................................................... 264 1. Radicalextensionsofafield ....................................................... 264 Extendingafieldbyaroot Complexroots 2. Thesymmetriesoftherootsofapolynomial ....................................... 270 3. Galoisgroups .................................................................... 273 Isomorphismsoffieldextensionsthatpermutetheroots Solvingpolynomialsbyradicals 4. “Solvable”groups.................................................................279 5. TheTheoremofAbelandRuffini..................................................283 A“reverse-Lagrange”Theorem Wecannotsolvethequinticbyradicals 6. TheFundamentalTheoremofAlgebra.............................................289 BackgroundfromCalculus Somemorealgebra ProofoftheFundamentalTheorem 10. Factorization.....................................................................294 1. Thelinkbetweenfactoringandideals.............................................294 2. UniqueFactorizationdomains....................................................297 3. FiniteFieldsI ................................................................... 300 Thecharacteristicofaring Example Mainresult iv 4. FinitefieldsII...................................................................307 5. Extendingaringbyaroot........................................................313 6. Polynomialfactorizationinfinitefields............................................316 Distinctdegreefactorization. Equaldegreefactorization Squarefreefactorization 7. Factoringintegerpolynomials....................................................324 Onebigirreducible. Severalsmallprimes. 11. Roots of multivariate polyomials..................................................327 1. Gaussianelimination ............................................................ 328 2. Monomialorderings.............................................................334 3. Matrixrepresentationsofmonomialorderings.....................................341 4. ThestructureofaGröbnerbasis..................................................344 5. Buchberger’salgorithm .......................................................... 354 6. Nullstellensatz .................................................................. 363 7. Elementaryapplications ......................................................... 365 12. Advanced methods of computing Gröbner bases .................................. 370 1. TheGebauer-Mölleralgorithm...................................................370 2. TheF4algorithm................................................................379 3. Signature-basedalgorithmstocomputeaGröbnerbasis.............................384 Part III. Appendices Where can I go from here? ........................................................... 393 Advancedgrouptheory ............................................................ 393 Advancedringtheory..............................................................393 Applications.......................................................................393 Hints to Exercises....................................................................394 HintstoChapter0.................................................................394 HintstoChapter1.................................................................394 HintstoChapter2.................................................................395 HintstoChapter3.................................................................396 HintstoChapter4.................................................................397 HintstoChapter5.................................................................398 HintstoChapter6.................................................................398 HintstoChapter7.................................................................399 HintstoChapter8.................................................................400 HintstoChapter10................................................................400 HintstoChapter11................................................................401 Index................................................................................402 References ........................................................................... 406 v 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 [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(α) fieldextensionofFbyal pha Frac(R) thesetoffractionsofacommutativering R F thesetofallfunctionsmapping S toitself S 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 (H) thenormalizerofasubgroup H ofG G N { } thenaturalnumbers 0,1,2,... N+ positiveintegers Ω the nthrootsofunity;thatis,allrootsofthepolynomial xn−1 n ord(x) theorderof x P(S) thepowersetof S Q thegroupofquaternions 8 R/A for R a ring and Aan ideal subring of R, R/Ais the quotient ring of R with respecttoA
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