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1、GALOISTHEORY1.IntroductionGaloistheorygrowsoutofattemptstosolvepolynomialequations.Let’shaveabriefhistoryofthisimportantproblem.(1)Quadraticequations.Recallhowonesolvesthequadraticequationx2+bx+c=0.We‘completethesquare’:bb2(x+)2+c−=0,24andsolveforx:x=−b+rb2−c.24Thiswasknownt
2、otheBabylonians(400BC,algorithmic),toEuclid(300BC,geometric),andtoBrahmagupta(6thcenturyAD,allowingnegativequantities,usinglettersforunknowns).(2)Cubicequations.Wenextconsideracubicequationx3+bx2+cx+d=0.Thiswasfirstsolved(atleastinspecialcases)bydalFerroin1515.Hekepthismetho
3、dssecretfor11yearsbeforepassinghisknowledgetohisstudentFior.By1535Tartagliahada‘general’solution,anddefeatedFiorinpubliccompetition.CardanoconvincedTartagliatodivulgehissolution,andbreakinganoathofsecrecy,publisheditinhisvolumeArsMagna.Thegeneralequationabovecanbereducedtoth
4、ecasey3+my=n(why?).ThenCardano’sformulaisssrr3n2m3n3n2m3ny=++−+−.42724272Remarkably,negativenumberswerenotunderstoodatthetime;theformula‘makessense’whenmandnarenonnegative.(3)Quarticequations.TheseweresolvedbyCardano’sstudentFerrari.(4)Quinticequations.Abelprovedin1824thatth
5、ereisnogeneralformulaforxsatisfyingtheequationx5+bx4+cx3+dx2+ex+f=0√intermsofb,c,d,e,fcombinedusing+,−,×,÷,n.Galois(1811-1312)wasthepioneerinthisfield.Heshowedthatsolutionsofapolynomialequationaregovernedbyagroup.Toillustratethis,letusconsiderthepolynomialx4=2.√Onesolutionis
6、thepositiverealfourthrootα=42of2.Theotherthreearethenβ√andδ=−i.42.Letuslistsomeequationsthatthesefourrootssatisfy:√√=i.42,γ=−42,α+γ=0,αβγδ=−2,αβ−γδ=0,....Whathappensifweswapαandγinthislist?Wegetγ+α=0,γβαδ=−2,γβ−αδ=0,...,whichareallstillvalidequations.Ifweperformthepermutati
7、onα7→β7→γ7→δ7→αtotheoriginallist,weobtainβ+δ=0,βγδα=−2,βγ−δα=0,...,1whichagainarealltrue.Webegintogettheimpressionthatanythinggoes.Butswappingαandβintheoriginallistgivesβ+γ=0,βαγδ=−2,βα−γδ=0,...,andthefirstoftheseequationsisfalse.Thesetofpermutationsofthesolutionsα,β,γ,δwhic
8、hpreservethevalidityofallpolynomialequationsformsagroup,theGaloisgroupofthe
