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LEADER: 10146cam 2200961Ii 4500
001 ocn893117743
003 OCoLC
005 20220521211933.0
008 141016s2014 sz o 000 0 eng d
006 m o d
007 cr cnu---unuuu
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019 $a897433528$a1005778844$a1011846516$a1022048097$a1048138399$a1066629552$a1086451938$a1111065733$a1112522872$a1113158753
020 $a9783319100944$q(electronic bk.)
020 $a3319100947$q(electronic bk.)
020 $a3319100939$q(print)
020 $a9783319100937$q(print)
020 $z9783319100937
024 7 $a10.1007/978-3-319-10094-4$2doi
035 $a(OCoLC)893117743$z(OCoLC)897433528$z(OCoLC)1005778844$z(OCoLC)1011846516$z(OCoLC)1022048097$z(OCoLC)1048138399$z(OCoLC)1066629552$z(OCoLC)1086451938$z(OCoLC)1111065733$z(OCoLC)1112522872$z(OCoLC)1113158753
037 $bSpringer
050 4 $aQA161.E95
072 7 $aMAT$x002040$2bisacsh
072 7 $aPBH$2bicssc
082 04 $a512.922$223
100 1 $aBilu, Yuri F.,$eauthor.
245 10 $aProblem of Catalan /$cYuri F. Bilu, Yann Bugeaud, Maurice Mignotte.
264 1 $aSwitzerland :$bSpringer,$c2014.
300 $a1 online resource
336 $atext$btxt$2rdacontent
337 $acomputer$bc$2rdamedia
338 $aonline resource$bcr$2rdacarrier
347 $atext file$bPDF$2rda
588 0 $aOnline resource; title from PDF title page (EBSCO, viewed October 16, 2014).
505 0 $6880-01$aAn Historical Account -- Even Exponents -- Cassels' Relations -- Cyclotomic Fields -- Dirichlet L-Series and Class Number Formulas -- Higher Divisibility Theorems -- Gauss Sums and Stickelberger's Theorem -- Mihăilescu?s Ideal -- The Real Part of Mihăilescu?s Ideal -- Cyclotomic units -- Selmer Group and Proof of Catalan's Conjecture -- The Theorem of Thaine -- Baker's Method and Tijdeman's Argument -- Appendix A: Number Fields -- Appendix B: Heights -- Appendix C: Commutative Rings, Modules, Semi-Simplicity -- Appendix D: Group Rings and Characters -- Appendix E: Reduction and Torsion of Finite G-Modules -- Appendix F: Radical Extensions.
520 $aIn 1842 the Belgian mathematician Eugène Charles Catalan asked whether 8 and 9 are the only consecutive pure powers of non-zero integers. 160 years after, the question was answered affirmatively by the Swiss mathematician of Romanian origin Preda Mihăilescu. In this book we give a complete and (almost) self-contained exposition of Mihăilescu?s work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume very modest background: a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.
650 0 $aConsecutive powers (Algebra)
650 0 $aProblem solving.
650 2 $aProblem Solving
650 6 $aPuissances consécutives (Algèbre)
650 6 $aRésolution de problème.
650 7 $aMATHEMATICS$xAlgebra$xIntermediate.$2bisacsh
650 7 $aConsecutive powers (Algebra)$2fast$0(OCoLC)fst00875466
650 7 $aProblem solving.$2fast$0(OCoLC)fst01077890
653 00 $awiskunde
653 00 $amathematics
653 00 $agetallenleer
653 00 $anumber theory
653 00 $aalgebra
653 10 $aMathematics (General)
653 10 $aWiskunde (algemeen)
655 0 $aElectronic books.
655 4 $aElectronic books.
700 1 $aBugeaud, Yann,$d1971-$eauthor.
700 1 $aMignotte, Maurice,$eauthor.
776 08 $iPrinted edition:$z9783319100937
856 40 $3ProQuest Ebook Central$uhttps://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=1965396
856 40 $3ebrary$uhttp://site.ebrary.com/id/10953317
856 40 $3EBSCOhost$uhttps://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=864756
856 40 $3SpringerLink$uhttps://doi.org/10.1007/978-3-319-10094-4
856 40 $3SpringerLink$uhttps://link.springer.com/book/10.1007/978-3-319-10094-4
856 40 $3SpringerLink$uhttps://link.springer.com/book/10.1007/978-3-319-10093-7
856 40 $3Scholars Portal$uhttp://books.scholarsportal.info/viewdoc.html?id=/ebooks/ebooks3/springer/2014-12-11/1/9783319100944
856 40 $3Scholars Portal Books$uhttp://www.library.yorku.ca/e/resolver/id/2595227
856 40 $3SpringerLink$uhttp://www.library.yorku.ca/e/resolver/id/2595228
856 4 $uhttps://discover.gcu.ac.uk/discovery/openurl?institution=44GLCU_INST&vid=44GLCU_INST:44GLCU_VU2&?u.ignore_date_coverage=true&rft.mms_id=991002483687803836$p5370671240003836$xStGlGCU
880 00 $6505-01/(S$gMachine generated contents note:$g1.$tHistorical Account --$g1.1.$tCatalan's Note Extraite --$g1.2.$tParticular Cases --$g1.3.$tCassels' Relations --$g1.4.$tAnalysis: Logarithmic Forms --$g1.5.$tAlgebra: Cyclotomic Fields --$g1.6.$tNumerical Results --$g1.7.$tFinal Attack --$g2.$tEven Exponents --$g2.1.$tEquation xp = y2 + 1 --$g2.2.$tUnits of Real Quadratic Rings --$g2.3.$tEquation x2 -- y" = 1 with Q [≥] 5 --$g2.4.$tCubic Field Q(3[√]2) --$g2.5.$tEquation x2 -- y3 = 1 --$g3.$tCassels' Relations --$g3.1.$tCassels' Divisibility Theorem and Cassels' Relations --$g3.2.$tBinomial Power Series --$g3.3.$tProof of the Divisibility Theorem --$g3.4.$tHyyro's Lower Bounds --$g4.$tCyclotomic Fields --$g4.1.$tDegree and Galois Group --$g4.2.$tIntegral Basis and Discriminant --$g4.3.$tDecomposition of Primes --$g4.4.$tUnits --$g4.5.$tReal Cyclotomic Field and the Class Group --$g4.6.$tCyclotomic Extensions of Number Fields --$g4.7.$tGeneral Cyclotomic Fields --$g5.$tDirichlet L-Series and Class Number Formulas --$g5.1.$tDirichlet Characters and L-Series --$g5.2.$tDedekind ζ-Function of the Cyclotomic Field --$g5.3.$tCalculating L(1, χ) for χ [≠] 1 --$g5.4.$tClass Number Formulas --$g5.5.$tComposite Moduli --$g6.$tHigher Divisibility Theorems --$g6.1.$tMost Important Lemma --$g6.2.$tInkeri's Divisibility Theorem --$g6.3.$tDeviation: Catalan's Problem with Exponent 3 --$g6.4.$tGroup Ring --$g6.5.$tStickelberger, Mihailescu, and Wieferich --$g7.$tGauss Sums and Stickelberger's Theorem --$g7.1.$tStickelberger's Ideal and Stickelberger's Theorem --$g7.2.$tGauss Sums --$g7.3.$tMultiplicative Combinations of Gauss Sums --$g7.4.$tPrime Decomposition of a Gauss Sum --$g7.5.$tProof of Stickelberger's Theorem --$g7.6.$tKummer's Basis --$g7.7.$tReal and the Relative Part of Stickelberger's Ideal --$g7.8.$tProof of Iwasawa's Class Number Formula --$g8.$tMihailescu's Ideal --$g8.1.$tDefinitions and Main Theorems --$g8.2.$tAlgebraic Number (x -- ζ) --$g8.3.$tqth Root of (x -- ζ) --$g8.4.$tProof of Theorem 8.2 --$g8.5.$tProof of Theorem 8.4 --$g8.6.$tApplication to Catalan's Problem I: Divisibility of the Class Number --$g8.7.$tApplication to Catalan's Problem II: Mihailescu's Ideal vs Stickelberger's Ideal --$g8.8.$tOn the Real Part of Mihailescu's Ideal --$g9.$tReal Part of Mihailescu's Ideal --$g9.1.$tMain Theorem --$g9.2.$tProducts of Binomial Power Series --$g9.3.$tMihailescu's Series (1 + ζ T)/q --$g9.4.$tProof of Theorem 9.2 --$g10.$tCyclotomic Units --$g10.1.$tCirculant Determinant --$g10.2.$tCyclotomic Units --$g11.$tSelmer Group and Proof of Catalan's Conjecture --$g11.1.$tSelmer Group --$g11.2.$tSelmer Group as Galois Module --$g11.3.$tUnits as Galois Module --$g11.4.$tq-Primary Cyclotomic Units --$g11.5.$tProof of Theorem 11.5 --$g12.$tTheorem of Thaine --$g12.1.$tIntroduction --$g12.2.$tPreparations --$g12.3.$tProof of Theorem 12.2 --$g12.4.$tReduction of a Multiplicative Group Modulo a Prime Ideal --$g12.5.$tReduction of a Multiplicative Group Modulo a Prime Number and Proof of Theorem 12.3 --$g13.$tBaker's Method and Tijdeman's Argument --$g13.1.$tIntroduction: Thue, Gelfond, and Baker --$g13.2.$tHeights in Finitely Generated Groups --$g13.3.$tAlmost nth Powers --$g13.4.$tEffective Analysis of Classical Diophantine Equations --$g13.5.$tTheorem of Schinzel and Tijdeman and the Equation of Pillai --$g13.6.$tTijdeman's Argument --$gAppendix$tA Number Fields --$gA.1.$tEmbeddings, Integral Bases, and Discriminant --$gA.2.$tUnits, Regulator --$gA.3.$tIdeals, Factorization --$gA.4.$tNorm of an Ideal --$gA.5.$tIdeal Classes, the Class Group --$gA.6.$tPrime Ideals, Ramification --$gA.7.$tGalois Extensions --$gA.8.$tValuations --$gA.9.$tDedekind ζ-Function --$gA.10.$tChebotarev Density Theorem --$gA.11.$tHilbert Class Field --$gAppendix B$tHeights --$gAppendix C$tCommutative Rings, Modules, and Semi-simplicity --$gC.1.$tCyclic Modules --$gC.2.$tFinitely Generated Modules --$gC.3.$tSemi-simple Modules --$gC.4.$tSemi-simple Rings --$gC.5.$t"Dual" Module --$gAppendix D$tGroup Rings and Characters --$gD.1.$tWeight Function and the Norm Element --$gD.2.$tCharacters of a Finite Abelian Group --$gD.3.$tConjugate Characters --$gD.4.$tSemi-simplicity of the Group Ring --$gD.5.$tIdempotents --$gAppendix E$tReduction and Torsion of Finite G -Modules --$gE.1.$tTelescopic Rings --$gE.2.$tProducts of Telescopic Rings --$gE.3.$tElementary Divisors and Finitely Generated Modules --$gE.4.$tReduction and Torsion --$gAppendix F$tRadical Extensions --$gF.1.$tField Generated by a Single Root --$gF.2.$tKummer's Theory --$gF.3.$tGeneral Radical Extensions --$gF.4.$tEquivariant Kummer's Theory.
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