beejaganita samkhya sidhaantham

poornnasamkhyakal, yukthisahamaya samkhyakal, avayude saamaanyavalkkaranangal enniva padikkaan amoortha beejaganithatthinte saankethikavidyakal upayogikkunna samkhya sidhaanthathinte oru shaakhayaanu beejaganita samkhya sidhaantham. beejaganita samkhyakal, avayude poornnasamkhyakalude valayangal, parimithamaaya feeldukal, function feeldukal thudangiya beejaganita vasthukkalude gunangalude adisthaanathilaanu samkhyaa-saidhaanthika chodyangal prakadippikkunnathu. oru valayam athulyamaaya factorisation amgeekarikkunnundo, aadarshangalude swabhaavam, feeldukalude galois grouppukal enniva polulla eee savisheshathakalkk diophantine samavakyangalkkulla parihaarangalude nilanilpp pole samkhyaa sidhaanthathile praadhamika praadhaanyamulla chodyangal pariharikkan kazhiyum.
beejaganita samkhyaasampradaayathinte charithram.
diophantus.
beejaganita samkhyaa sidhaanthathinte thudakkam moonnaam noottaandile alexandrian ganithashaasthranjanaaya diophantasinte perilulla diophantine samavakyangalil ninnu kandethaam, avayekkurichu padikkukayum chilatharam diophantine samavaakyangalude parihaarathinulla reethikal vikasippikkukayum cheythu. x", "y" ennee rund poornnasamkhyakalum avayude koottavum avayude chathurangalude koottavum yathaakramam "a", "b" ennee rund samkhyakalkku thulyamaaya reethiyil kandethuka ennathaanu oru saadhaarana diophantine prashnam.
diophantine samavaakyangal aayirakkanakkin varshangalayi padikkappettittund. udaaharanathinu, chathuraakrithiyilulla diophantine samavaakyam "x" 2 + "y" 2 = "z" 2-nulla pariharangal nalkunnathu pythagorian trippilukalaanu, yadhaarthathil babyloniakkaar () aanu ithu pariharichathu. 26 'x' + 65 'y' = 13 polulla leanier diophantine samavakyangalkkulla pariharangal euclidean algoritam (c. bii. sii. anchaam noottaand).
diophantasinte pradhana kruthi "ganithashaasthram" aayirunnu, athil oru bhagam maathrame avasheshichittulloo.
fermet.
1637-ll piyari dii fermet aanu fermattinte avasana sidhaantham aadyamaayi oohichathu, "ganitha" thinte oru pakarppinte maarjinil addeham avakaashappettath athinte maarjinil yojikkan kazhiyaathathra valuthaanu ennathinu oru thelivu undennaanu. 358 varshathinide ennamatta ganithashaasthranjarude sramangal undaayirunnittum 1995 vare vijayakaramaya thelivukalonnum prasidheekarichirunnil. pariharikkappedaatha prashnam pathonpathaam noottaandil beejaganita samkhyaa sidhaanthathinteyum irupathaam noottaandil modularity sidhaanthathinteyum vikaasathe uthejippichu.
gass.
beejaganita samkhyaa sidhaanthathinte sthaapaka kruthikalilonnaaya disquisisance ganithashaasthram (laattin "ganithashaasthra anweshanangal") 1798ll kaal friedrich gass ezhuthiya samkhyaa sidhaanthathinte paatapusthakamaanu. anweshanangal" prasidheekarikkunnathinnum, samkhyaa sidhaanthathil otappetta sidhaanthangaludeyum anumaanangaludeyum oru shekharam ulppettirunnu. gass thante mungaamikalude kruthikaleyum thante swantham yadhaartha kruthikaleyum oru chittayaya chattakkoodilekku konduvannu, vidavukal nirachu, sariyallaatha thelivukal sariyaakkukayum vishayam niravadhi vazhikalil vipuleekarikkukayum cheythu.
pathonpathaam noottaandile mattu europian ganithashaasthranjaraaya ernst kummer, peettar gustav lejyoon dirichlett, richard dedekind ennivarude pravartthanangalude thudakkamaayirunnu "discusitions". gaus nalkia vyaakhyaanangalil palathum phalatthil addehathinte swantham kooduthal gaveshana prakhyaapanangalaan, avayil chilath prasidheekarikkappedaathe thanne thudarunnu. avaa addehathinte samakalikarkk prathyekichum nigoodamaayi thonniyirikkanam; prathyekichu ell-functionukaludeyum sankeernamaaya gunanathinteyum sidhaanthangalude anukkal adangiyirikkunnuvennu namukk ippol vaayikkam.
diriclet.
diriclet 1838lum rund prabandhangalil peettar gustav lejyoon dirichlett quadratic roopangalkkulla (pinneedu addehathinte vidyaarthi leopold chronecker parishkaricha) phastt klaas nambar formula theliyichu. manushyante ettavum kooduthal budhishakthiye sparsikkunna" phalam ennu jacoby vilicha eee soothravaakyam kooduthal pothuvaaya samkhya feeldukalumaayi bandhappett samanamaaya falangalkku vazhiyorukki. quadratic feeldukalude yoonit grouppinte gadanayekkurichulla thante gaveshanathe adisthaanamaakki, beejaganita samkhya sidhaanthathile adisthaana falamaaya diriclet yoonit sidhaantham addeham theliyichu.
diophantine ekadesharekhayile oru sidhaanthathinte thelivaayi addeham aadyam praavinte dwaaram thathwam, oru adisthaana ennal vaadam, upayogichu, pinneedu diriklettinte ekadesharekha sidhaantham addehathinte perilaanu. fermaattinte avasana sidhaanthathilekkum "n" = 5, "n" = 14 ennee kesukal theliyikkunnathilekkum dvikhaatha paraspara vinimaya niyamathilekkum addeham pradhana sambhavanakal prasidheekarichu. addeham aadya phalangal kandethiya diriclet devisor prashnam, mattu gaveshakar pinneedu nalkiyittum samkhyaasaahithyathil pariharikkappedaatha oru prasnamaanu.
samshayikkuka.
lejyoon dirichlettinte kruthikalekkurichulla richard didekindinte padanamaanu beejaganita samkhyaasampradaayangaleyum aadarshangaleyum kurichulla pilkkala padanatthilekku addehathe nayichathu. 1863-ll addeham lejyoon diriklettinte samkhyaa sidhaanthathekkurichulla prabhashanangal "worlessungen bubber sahlentherori" ("samkhyaa sidhaanthathekkurichulla prabhashanangal") enna peril prasidheekarichu, athinekkurichu ingane ezhuthiyittundu
vorlessengen"-nte 1879,1894 pathippukalil ring sidhaanthathinu anuyojyavum adisthaanaparavumaaya oru ashayam avatharippikkunna anubandhangal ulppedunnu. (hilbert pinneedu avatharippicha "ring" enna vaakk didekindinte kruthiyil kaanapedunnilla.) dedeckind oru idealine nirvachichathu poornnasankhya gunakangalulla bahupada samavaakyangale thripthippeduthunna beejaganita poornnasamkhyakal chernna oru koottam samkhyakalude upavibhagamaanu. eee ashayam hilbertinteyum prathyekichu emmy notherinteyum kaikalil kooduthal vikasichu. fermaattinte avasana sidhaantham theliyikkaanulla 1843 leeii kumaarinte shramathinte bhaagamaayi roopappeduthiya ernst edward kumaarinte ideal nambarukale idealukal saamaanyavalkkarikkunnu.
hilbert.
david hilbert 1897-ll thante grandhamaaya "salberict" (aksharaarthatthil "samkhyakalekkurichulla report") upayogichu beejaganita samkhyaa sidhaantha mekhalaye ekeekarichu. 1770ll warring roopeekaricha oru ganyamaaya samkhyaasaahithyaprashnam addeham pariharichu. parimitataa sidhaanthathilennapole, utharangal nalkunnathinulla oru samvidaanam nalkunnathinupakaram prashnathinu pariharangal undaayirikkanamennu kaanikkunna oru asthithvaprashnam addeham upayogichu. eee vishayathil addehathinu kooduthal prasidheekarikkaan kuravaayirunnu; ennaal oru vidyaarthiyude prabandhatthil hilbert modular roopangalude aavirbhavam ardhamaakkunnathu addehathinte peru oru pradhana mekhalayumaayi kooduthal bandhappettirikkunnu ennaanu.
klaas feeld sidhaanthathekkurichu addeham niravadhi oohaapohangal nadatthi. eee ashayangal valare swaadheenamullavayaayirunnu, hilbert klaas feeldinteyum praadeshika klaas feeld sidhaanthathinteyum hilbert chinnathinteyum perukalil addehathinte swantham sambhavana nilanilkkunnu. 1930 aayappozhekkum, teeji thakagiyude pravarthanatthinu sesham phalangal kooduthalum theliyikkapettu.
artin.
emil artin niravadhi prabandhangalil (1924; 1927; 1930) artin resiprosity niyamam sthaapichu. aagola klaas feeld sidhaanthathinte kendra bhagamaaya samkhyaasampradaayathile oru pothu sidhaanthamaanu eee niyamam. resiprosity loow" enna padam quadratic resiprosity niyamavum aisansteenteyum kammirinteyum resiprosity niyamangalum muthal norm chinnathinaayulla hilbertinte uthppanna soothravaakyam vare saamaanyavalkkaricha kooduthal vyakthamaaya nambar saidhaanthika prasthaavanakalude oru neenda variye soochippikkunnu. aarttinte phalam hilbertinte onpathaamathe prashnathinu bhagikamaaya pariharam nalki.
aadhunika sidhaantham.
1955 oode japanese ganithashaasthranjaraaya goro shimura yudaaka thaniyamayum ganithashaasthrathinte rund shaakhakal, deerghavrithaakrithiyilulla valavukal, modular roopangal enniva thammil saadhyamaaya oru bandham nireekshichu. thathfalamaayundaakunna modularity sidhaantham (akkaalat taniama-shimura anumaanam ennariyappettirunnu) oro deerghavrithaakrithiyilulla vakravum modular aanennu prasthaavikkunnu, athaayath athine oru advitheeya modular roopavumaayi bandhappedutthaam.
thudakkathil ithu asaadhyamo valare oohakkachavadaparamo aanennu thallikkalanjirunnu, nambar saidhaanthikanaaya andrey weyl ithine pinthunaykkunna thelivukal kandethiyappol kooduthal gauravamaayi eduthuvenkilum thelivukalonnumilla; thalphalamaayi "athisayakaramaaya" anumaanam palappozhum taniama-shimura-weyl anumaanam ennariyappettu. thelivo nirasikkalo aavasyamulla pradhana anumaanangalude oru pattikayaya langlands prograaminte bhaagamaayi ithu maari.
1993 muthal 1994 vare, andrew wiles semistable eliptik karvukalkkulla modularity sidhaanthathinte oru thelivu nalki, ithu ribettinte sidhaanthavumaayi chernnu fermaattinte avasana sidhaanthathinu oru thelivu nalki. akkaalathe mikkavarum ellah ganithashaasthranjarum fermattinte avasana sidhaanthavum modularity sidhaanthavum theliyikkaan asaadhyamo phalatthil asaadhyamo aanennu munbu kanakkaakkiyirunnu, ettavum athyaadhunikamaaya sambhavavikasangal polum kanakkiledukkumbol. 1993 juunil wiles aadyamaayi thante thelivu oru pathippil prakhyaapichu, athu oru pradhana ghattathil gurutharamaaya vidavu undennu taamasiyaathe amgeekarikkappettu. richard taylorumaayi sahakarichu wiles eee thelivu sariyaakkukayum 1994 septambaril puratthirakkiya anthimavum vyaapakamaayi amgeekarikkappettathumaaya pathipp 1995 ll aupachaarikamaayi prasidheekarikkukayum cheythu. eee thelivu beejaganita jyamithiyil ninnum samkhyaa sidhaanthathil ninnum niravadhi saankethikavidyakal upayogikkunnu, koodaathe ganithashaasthrathinte eee shaakhakalil niravadhi prathyaaghaathangalundu. skeemukalude vibhaagavum ivasava sidhaanthavum polulla aadhunika beejaganita jyaamithiyude staanderd nirmmaanangalum fermattinu labhyamallaatha mattu 20-aamm noottaandile saankethikavidyakalum ithu upayogikkunnu.
adisthaana ashayangal.
athulyamaaya faaktaraiseshante parajayam.
poornnasamkhyakalude valayathinte oru pradhana savisheshatha, athu ganithashaasthrathinte adisthaana sidhaanthathe thripthippeduthunnu ennathaanu, oro (positteev) poornnasamkhyaykkum prime nambarukalude gunanafalamaayi oru factorisation undu, eee factorisation gadakangalude kramam vare savisheshamaan. oru beejaganita samkhyaasampradaayathile poornnasamkhyakalude valayathil ithu inii sariyaayirikkilla.
oru uthppannathe vibhajikkukayaanenkil, athu gadakangalilonnine allengil gadakangalilonnine vibhajikkunna oru gadakamaanu "prime elament". eee property poornnasamkhyakalile primalityumaayi adutha bandhamullathaan, kaaranam eee property thripthippeduthunna eathoru positteev poornnasamkhyayum onnukil oru prime nambaraanu. ennirunnaalum, ithu karsanamaayi durbalamaanu. udaaharanathinu, ithu oru prime nambaralla, kaaranam ithu negatteevu aanu, pakshe ithu oru prime gadakamaanu. prime moolakangalilekkulla factorisationukal anuvadaneeyamaanenkil, poornnasamkhyakalil polum, mattu factorisationukal undu
pothuve, gunana vipareethamulla oru samkhyaye ardhamaakkunna oru unitaanengil, oru prime gadakamaanenkil, athum oru prime gadakamaanu. associate" ennu parayappedunna aandu polulla samkhyakal. poornnasamkhyakalil, avibhaajyaghadakangalum associate aakunnu, ennaal ivayil onnu maathrame positteev aakunnulloo. prime nambarukal positteev aayirikkanamenn aavasyappedunnathinaal anubandha prime moolakangalude oru koottatthil ninnu oru savishesha ghadakam thiranjedukkunnu. k" yukthisahamaya samkhyakalallengil, posittivittikku analog illa. udaaharanathinu, gausiyan poornnasamkhyakalil, akkangalum avayum associate cheiyunnu, kaaranam randaamathethu aadyathethinte gunanafalamaanu, ennaal onnu mattonninekkaal canonical aanennu ottappeduthaan oru margavumilla. ithu polulla samavaakyangalilekku nayikkunnu
gadakangalude kramam vare factorisationukal advitheeyamaanennathu sariyallennu ithu theliyikkunnu. ekkaaranathaal, advitheeya factorisation domainukalil (yuu. eph. dii. ezz) upayogikkunna advitheeya faaktaraiseshante nirvachanam oraal sweekarikkunnu. oru yuu. eph. dii. yill, oru factorisationil sambhavikkunna pradhana gadakangal unitukalkkum avayude kramatthinum maathramaayi mathramaayirikkum pratheekshikkappedunnath.
ennirunnaalum, eee durbalamaaya nirvachanam undaayirunnittum, beejaganita samkhya feeldukalile poornnasamkhyakalude pala valayangalum advitheeyamaaya gadakavalkkaranam amgeekarikkunnilla. ideal klaas group ennu vilikkunna oru beejaganita thadassamundu. anuyojyamaaya klaas group nissaaramaayirikkumbol, mothiram oru yuu. eph. dii aanu. athu illengil, oru prime moolakavum irreducible moolakavum thammil oru vyathyaasamund. irreducible elament" ennathu onnukil allengil oru unitaaya oru moolakamaan. kooduthal gadakangal cheyyaan kazhiyaatha gadakangalaaniva. oo" yile oro gadakavum oru gadakavalkkaranam irreducible gadakangalaayi amgeekarikkunnu, pakshe athu onniladhikam gadakangal amgeekarichekkaam. kaaranam, ellah prime moolakangalum irreducible aanenkilum, chila irreducible moolakangal prime aayirikkilla. udaaharanathinu, mothiram pariganikkuka. eee valayathil, samkhyakal, avaa irreducible aanu. ithinartham eee samkhyaykku irreducible gadakangalaayi rund gadakavalkkaranangalundu ennaanu.
eee samavaakyam uthppannathe vibhajikkunnuvennu kaanikkunnu. oru prime moolakamaayirunnenkil, athu vibhajikkumaayirunnu allengil, pakshe athu cheiyunnilla, kaaranam vibhajicha ellah gadakangalum roopathilullathaanu. athupole, uthppannathe vibhajichu, ennaal eee gadakangalonnum svayam vibhajikkunnilla, athinaal avaa randum avibhaajyamalla. moolakangalkk thulyamakkaan kazhiyunna oru ardhavumillaathathinaal, athulyamaaya gadakavalkkaranam paraajayappedunnu. nirvachanathe durbalappeduthunnathiloodu savisheshatha nannakkaan kazhiyunna yoonittukalude saahacharyathil ninnu vyathyasthamayi, eee paraajayatthe marikadakkaan oru puthiya veekshanam aavasyamaanu.
prime idealukalilekkulla factorisation.
oru ideal aanenkil, ellaayppozhum oru factorisation undu.
oro soothravaakyavum _ 7 oru prime ideal aanenkil, eee padaprayogam gadakangalude kramam vare advitheeyamaanu. prathyekichum, orotta moolakam srishtikkunna pradhana ideal aanenkil ithu sariyaanu. oru pothu samkhya feeldinte poornnasamkhyakalude valayam advitheeyamaaya gadakavalkkaranam sweekarikkunna ettavum sakthamaaya ardhamaanithu. ring sidhaanthathinte bhashayil, poornnasamkhyakalude valayangal dedeckined domainukalaanennu athu parayunnu.
oru yuu. eph. dii. aayirikkumbol, oro prime idealum oru prime moolakam srishtikkunnu. allaathapaksham, prime gadakangalaal srishtikkappedaatha prime idealukalundu. udaaharanathinu, oru moolakam kondu srishtikkaan kazhiyaatha oru prime idealaanu ideal.
charitraparamaayi, aadarshangale prime aadarshangalaayi factoring cheyyuka enna aashayathin munbu ernst cummer aadarsha samkhyakal avatharippichu. iva oru ekstenshan feeldil kidakkunna samkhyakalaanu. eee ekstenshan feeld ippol hilbert klaas feeld ennariyappedunnu. pradhana ideal sidhaanthamanusarichu, oro prime idealum poornnasamkhyakalude valayathinte oru pradhana ideal srishtikkunnu. eee pradhana aidiyalinte janarettarine ideal nambar ennu vilikkunnu. cyclotomic feeldukalile advitheeya factorisation paraajayappedunnathinu pakaramaayi kummar iva upayogichu. iva oduvil aadarshangalude oru mungami avatharippikkanum aadarshangalude athulyamaaya gadakavalkkaranam theliyikkaanum richard didekinthine nayichu.
oru samkhya feeldile poornnasamkhyakalude valayathil prime aaya oru ideal oru valia samkhya feeldilekku neettumbol prime aakunnathil paraajayappettekkam. udaaharanathinu, prime nambarukal pariganikkuka. bandhapetta idealukal ringinte prime idealukalaanu. ennirunnaalum, eee ideal gausiyan poornnasamkhyakalilekku vyaapippikkumbol, athu primaayirikkaam allengil illaayirikkaam. udaaharanathinu, factorisation soochippikkunnathu
kaaranam, avaa srishtikkunna aadarshangalum avayum onnuthanneyaanennu shradhikkuka. gassian poornnasamkhyakalil eathu idealukal primaayi thudarunnu enna chodyathinulla poornnamaaya utharam rund squareukalude thukaye adisthaanamaakkiyulla fermet sidhaantham nalkunnu. oru ottasamkhyaykku, oru prime ideal aanenkil athu oru prime ideal allengil athu oru prime ideal aanennu ithu soochippikkunnu. ithu, ideal prime aanenna nireekshanathodoppam, gausiyan poornnasamkhyakalile prime idealukalude poornnamaaya vivaranam nalkunnu. eee lalithamaaya phalam poornnasamkhyakalude kooduthal pothuvaaya valayangalilekku saamaanyavalkkarikkunnath beejaganita samkhyaasampradaayathile oru adisthaana prasnamaanu. k" ennathu q-nte abelian extensionaayirikkum (athaayath abelian galois grouppilulla oru galois ekstenshan) klaas feeld sidhaantham eee lakshyam kaivarikkunnu.
anuyojyamaaya klaas group.
pradhaanamaayirikkaatha prime idealukal undengil maathram advitheeya factorisation paraajayappedunnu. prime idealukalude parajayam principal aayi alakkunna vasthuvine ideal klaas group ennu vilikkunnu. anuyojyamaaya klaas groupine nirvachikkunnathin beejaganita poornnasamkhyakalude oru valayathile idealukalude koottam vipuleekarikkendathundu, angane avaa oru group ghadana amgeekarikkunnu. idealukale fraakshanal idealukalaayi saamaanyavalkkarichaanu ithu cheiyunnathu. oru fraakshanal ideal ennathu oru additteev upavibhagamaanu, athinte gadakangalaal gunanathinu keezhil adaykkunnu, athaayath. ellah idealukalum fraakshanal idealukalaanu. fraakshanal idealukalaanengil, oru moolakathinteyum oru moolakathinteyum ellah uthppannangaludeyum koottavum oru fraakshanal idealaanu. eee pravartthanam poojyamallaatha bhinnasamkhyakalude koottatthe oru grooppaakkunnu. group identityaanu ideal, athinte vipareethamaanu (saamaanyavalkkaricha) ideal koshyanta
poojyamallaatha ellah bhinna idealukaludeyum grouppinte oru upavibhagam roopappeduthunna roopathilullava ennartham varunna pradhana bhinna idealukal. eee upavibhagam poojyamallaatha bhinnasamkhyakalude koottatthinte bhagamaanu anuyojyamaaya klaas group. rund fraakshanal idealukal, atharam oru ghadakam undengil maathram anuyojyamaaya klaas grouppinte ore gadakathe prathinidheekarikkunnu. athinaal, onnu mattonnineppole pradhaanamaayirikkunnathinno aduthaanengil anuyojyamaaya klaas group rund fraakshanal idealukale thulyamaakkunnu. anuyojyamaaya klaas groupine saadhaaranayaayi soochippikkunnu, allengil (beejaganita jyaamithiyile picard grouppumaayi avasaanathe notteshan athine thirichariyunnu).
klaas groupile gadakangalude ennathe "keey" yude klaas nambar ennu vilikkunnu. klaas nambar 2 aanu. ithinartham rund ideal klaasukal maathrameyulloo, principal fraakshanal idealukalude klaas, principal allatha fraakshanal aidiyalinte klaas.
anuyojyamaaya klaas groupinu divaisarukalude kaaryatthil mattoru vivaranamundu. samkhyakalude saadhyamaaya gadakavalkkaranangale prathinidheekarikkunna aupachaarika vasthukkalaaniva. avibhaajya idealukal srishtikkunna swatantra abelian grouppaanu divisar group. poojyamallaatha gadakangalil ninnu gunanam vare oru group homomorphisamundu. athu thripthikaramaanennu karuthuka
appol athu vibhajakamaayi nirvachikkappedunnu
kernel yoonittukalude grouppaanu, athesamayam cockernal anuyojyamaaya klaas grouppaanu. homological beejaganithatthinte bhashayil, abelian grouppukalude krithyamaaya oru kramamundennu ithu parayunnu (gunithamaayi ezhuthunnu),
yadhaarthavum sankeernavumaaya ulcherkkalukal.
chila nambar feeldukal, yadhaartha samkhyakalude subfieldukalaayi vyakthamaakkam. mattullavarkk, udaaharanathinu, kazhiyilla. amoorthamaayi, atharamoru specification oru feeld homomorphisavumaayi allengil yojikkunnu. ivaye yathaakramam yadhaartha ulcherkkalukalum sankeernna ulcherkkalukalum ennu vilikkunnu.
rund yadhaartha ulcherkkalukal amgeekarikkukayum ennaal sankeernamaaya ulcherkkalukalonnathil oru thikanja chathuramallattha oru yadhaartha chathuraakrithiyilulla feeld ennu vilikkappedunnu. yathaakramam athilekkum athilekkum ayaykkunna feeld homomorphisangalaanu iva. eratta, oru saangalppika chathuraakrithiyilulla feeld yadhaartha ulcherkkalukalonnum amgeekarikkunnilla, pakshe sankeernamaaya ulcherkkalukalude oru samyojitha jody amgeekarikkunnu. eee ulcherkkalukalilonnu ayaykkunnu, mattonn athinte sankeernamaaya samyojanathilekku ayaykkunnu.
parambaraagathamaayi, yadhaartha ulcherkkalukalude ennam soochippikkunnu, athesamayam sankeernamaaya ulcherkkalukalude samyojitha jodikalude ennam soochippikkunnu. keey" yude oppu jody aanu. ithu oru sidhaanthamaanu, evideyaanu digri.
ellah ulcherkkalukalum oresamayam pariganikkunnath oru function formula _ 13, allengil thulyamaaya formula _ 14 nirnnayikkunnu.
ithine minkovsky embedding ennu vilikkunnu.
sankeernamaaya samyojanam vazhi nischayichirikkunna codomaininte upa idam minkovsky space ennu vilikkunna alavilulla oru yadhaartha vekttar space aanu. mingovsky embedding feeld homomorphisangalaal nirvachikkappedunnathinaal, oru moolakatthinte moolakangalude gunanam mingovsky embeddingile diagonal matricks kondu gunikkunnathinodu yojikkunnu. minkovsky sthalathe dott uthppannam trace phoom formula _ 15-maayi poruthappedunnu.
minkovsky embeddingin keezhilulla chithram oru-dimensional laattis aanu. eee laattisinte adisthaanamaanengil, athu vivechanamaanu. vivechanam soochippikkunnathu allengil. chithratthinte covolium formula _ 16 aanu.
sthalangal.
moolyanirnnayathe adisthaanamaakkiyulla oru veekshanam sweekarikkunnathiloode yadhaarthavum sankeernavumaaya ulcherkkalukal pradhana aadarshangalude athe aditharayil sthaapikkaan kazhiyum. udaaharanathinu, poornnasamkhyakal pariganikkuka. saadhaarana kevala moolya pravarthanatthinu purame |·| : q → r, p-adic kevala moolya pravartthanangal undu |·|p: q → r, oro prime nambarinum "p" ennu nirvachichirikkunnu, ithu "p" kondu vibhajyatha alakkunnu. ivayellaam q (thulyatha vare) ll saadhyamaaya kevala moolya pravarthanangalaanennu austrovskyude sidhaantham prasthaavikkunnu. athinaal, q-nteyum avibhaajyasamkhyakaludeyum yadhaartha ulcherkkal vivarikkunnathinulla oru pothu bhaashayaan kevala moolyangal.
k" ennathil samboornna moolya pravartthanangalude oru thulyatha vibhagamaanu beejaganita samkhya feeldinte oru sthaanam. rund tharatthilulla sthalangalundu. o" yude oro prime ideal formulakkum 17-edic kevala moolyam enna oru soothravaakyam undu, koodaathe "p"-edic kevala moolyangal pole, athu vibhajanakshamatha alakkunnu. ivaye parimithamaaya sthalangal ennu vilikkunnu. k" yude yadhaarthamo sankeernamo aaya ulcherkalum r allengil c-leeii staanderd kevala moolya pravartthanavum upayogichaanu mattoru tharam sthalam vyakthamaakkunnath. iva ananthamaaya sthalangalaanu. oru sankeernamaaya ulcherkalum athinte samyojithavum thammil verthirichariyaan samboornna moolyangalkku kazhiyaathathinal, oru sankeernamaaya ulcherkalum athinte samyojithavum ore sthalam nirnnayikkunnu. athinaal, yadhaartha sthalangalum sankeernamaaya sthalangalum undu. sthalangal praimukale ulkkollunnathinaal, sthalangale chilappol primukal ennu vilikkunnu. ithu cheyyumbol, parimithamaaya sthalangale parimithamaaya avibhaajyaghadakangal ennum ananthamaaya sthalangale ananthamaaya avibhaajyaghadakangal ennum vilikkunnu. oru kevala moolyavumaayi bandhapetta moolyanirnnayamaanenkil, oraal pathivayi formula _ 19 ezhuthunnu, athaayath athu ananthamaaya sthalavum formula _ 20 ennathu oru parimithamaaya sthalavumaanennu ardhamaakkunnu.
feeldinte ellah sthalangalum orumichu pariganikkumbol nambar feeldinte adel ring srishtikkunnu. kevala moolyangal upayogichu labhyamaya ellah daattayum oresamayam tracuchaian adele ring oraale anuvadikkunnu. artin resiprosity niyamatthilennapole, oridathe perumaattam mattu sthalangalile perumaattatthe badhikkunna saahacharyangalil ithu ganyamaaya nettangal srishtikkunnu.
jyamitheeyamaayi ananthathayil sthaanangal.
vakrangalude function feeldukalil nilanilkkunna ananthathayilulla sthalangalkku oru jyaamitheeya saamyam undu. udaaharanathinu, formula _ 21 umm formula _ 22 umm minusamaarnnathum projective, beejaganita vakramayirikkatte. function feeld formula _ 23 nuu niravadhi kevala moolyangalo sthalangalo undu, oronnum vakrathile oru pooyintumaayi poruthappedunnu. formula _ 24 ennathu oru affine karvu formula _ 25-nte projective poorttheekaranamaanenkil, formula _ 26-leeii pooyintukal ananthathayilulla sthalangalumaayi poruthappedunnu. appol, eee pooyintukalilonnil formula 27 poortthiyaakkunnathu formula 28-aadiksinte oru analog nalkunnu.
udaaharanathinu, formula 29 aanenkil athinte function feeld formula 30-lake isomorphic aanu, avide formula 31 oru anishchithathvamaanu, feeld formula 27 formula 31-leeii bahupadangalude bhinnasamkhyakalude feeldaanu. appol, oru pooyint formula oru plays formula 34 aprathyakshamaakunna kramam allengil pooyint formula 28-ll polynomial formula 36-nte oru bhinnasamkhyayude oru dhruvathinte kramam alakkunnu. udaaharanathinu, formula 38 aanenkil, affine chart formula 39-ll ithu pooyint formula 40-maayi poruthappedunnu, moolyanirnnaya formula 41 formula 42-nte aprathyakshamakaanulla kramam alakkunnu. formula 44-ll formula 43-nte aprathyakshamakaanulla kramam. formula 41-leeii poorttheekaranatthinte function feeld formula 46 aanu, ithu 47-leeii power seerees formula power seerees formulayude feeld aanu, athinaal eee feeldinte formula 49 -
unitukal.
poornnasamkhyakalkku rund unitukal maathrameyulloo, oppam. poornnasamkhyakalude mattu valayangal kooduthal unitukal sweekarikkaam. gausiyan poornnasamkhyakalkku naalu yoonittukalund, munbathe randum athupole thanne. aisanstain poornnasamkhyakalkku aarr yoonittukalund. yadhaartha chathurasankhya feeldukalile poornnasamkhyakalkku ananthamaaya niravadhi yoonittukalund. udaaharanathinu, oro shakthiyum oru yoonittaan, eee shakthikalellaam vyathyasthamaan.
pothuve, soochippicha yoonittukalude group, parimithamayi srishtikkappetta abelian grouppaanu. parimithamayi srishtikkappetta abelian grouppukalude adisthaana sidhaantham soochippikkunnathu ithu oru tortion bhaagathinteyum swatantra bhaagathinteyum nerittulla thukayaanu ennaanu. oru samkhyaasampradaayathinte pashchaathalatthil ithu punarvyaakhyaanikkumbol, tortion bhagam aikyathinte verukal ulkkollunnu. eee group chaakrikamaan. swathanthrabhaagathe vivarikkunnath diriklettinte yoonit sidhaanthamaanu. swatantra bhaagatthinte rank aanennu eee sidhaantham parayunnu. udaaharanathinu, swatantra bhaagatthinte rank poojyamaaya feeldukalum saangalppika quadratic feeldukalum maathramaan. k"/q enna galois groupinu oru galois modyoolaayi "o" × ̃zq-nte ghadana nalkunna kooduthal krithyamaaya prasthavana saadhyamaan.
yoonit grouppinte swatantra bhagam ananthamaaya sthalangal upayogichu padikkaam. pravartthanam pariganikkuka
ananthamaaya sthalangalil vyathyaasappedunna edavum |·|"v" ennathu athumaayi bandhapetta samboornna moolyamaanu. oru yadhaartha vekttar space muthal oru homomorphisamaanu function. eee laattisinte chithram formula _ 51 nirvachicha hyperplanil vyaapichukidakkunna oru laattis aanennu kaanikkan kazhiyum, eee laattisinte covolium aanu nambar feeldinte regulator. adele valayathil pravartthikkunnathiloode saadhyamaaya lalithavalkkaranangalilonn, eee laattis, ideal klaas group ennivayude bhaagaphalathe vivarikkunna ideal klaas group enna orotta vasthu maathrameyulloo ennathaanu.
seetta function.
oru samkhya feeldinte dedeckind seetta function, riman seetta functionode saamyamullathaan, ithu prime idealukalude swabhaavam vivarikkunna oru vishakalana vasthuvaan. oro diriclet pratheekatthinum oru ghadakam ulla didiclind seetta functionukal diriclet ell-functionukalude ulppannangalaanu. nissaramaaya aksharam riman seetta functionumaayi poruthappedunnu. oru galois ekstenshan eppozhaayirikkum, dedeckind seetta function galois grouppinte pathiv praathinidhyatthinte artin ell-functionaanu, koodaathe galois grouppinte irreducible artin praathinidhyangalude kaaryatthil ithinu oru factorisation undu.
klaas nambar formula mukalil vivaricha mattu inevarantukalumaayi seetta function bandhappettirikkunnu.
praadeshika mekhalakal.
w" enna sthalathu "k" enna nambar feeld poortthiyaakkunnathu oru poornna feeld nalkunnu. moolyanirnnayam archemedian aanenkil, oraalkk r allengil c labikkum, athu archemedian allaathathum yukthikalude oru prime "p" nuu mukalilaanenkil, oraalkk oru parimithamaaya ekstenshan formula _ 52 labikkum, parimithamaaya avashishta feeldullah poornnavum vyathirikthavumaya moolyamulla feeld. eee prakriya feeldinte ganithathe lalithamaakkukayum prashnangalekkurichulla praadeshika patanam anuvadikkukayum cheiyunnu. udaaharanathinu, chronecker-weber sidhaantham analog local stettmentil ninnu eluppathil manassilakkaan kazhiyum. praadeshika mekhalakalekkurichulla padanatthinu pinnile thathwachintha pradhaanamaayum jyaamitheeya reethikalil ninnu prachodanam ulkkondathaanu. beejaganita jyamithiyil, oru maximal idealilekku praadeshikavalkkarichukonu oru ghattathil praadeshikamaayi enangale padikkunnathu saadhaaranamaanu. praadeshika daatta orumichu cherkkunnathiloode aagola vivarangal veendedukkaan kazhiyum. beejaganita samkhyaasampradaayathil eee aathmaav sweekarikkappettirikkunnu. oru samkhyaasampradeshathile beejaganita poornnasamkhyakalude valayathil oru prime nalkiyaal, aaa prime feeld praadeshikamaayi padikkunnathu abhikaamyamaanu. athinaal, oraal beejaganita poornnasamkhyakalude valayam aaa primilekku praadeshikavalkkarikkum thudarnnu jyaamithiyude aathmaavil bhinnasankhya feeld poortthiyaakkukayum cheiyunnu.
pradhana phalangal.
klaas grouppinte parimithatha.
beejaganita samkhya sidhaanthathile classical falangalilonnu oru beejaganita samkhya feeld "k"-nte anuyojyamaaya klaas group parimithamaanu ennathaanu. oru nischitha positteev poornnasankhya peju 78-ll thaazhe normullah parimithamaaya niravadhi avibhaajya idealukal maathrame ulloo ennathinaal ithu minkovsky sidhaanthathinte anantharafalamaanu. klaas grouppinte kramatthe klaas nambar ennu vilikkunnu, palappozhum "echu" enna aksharam kondu soochippikkunnu.
diriklettinte yoonit sidhaantham.
diriklettinte yoonit sidhaantham poornnasamkhyakalude valayathinte "o" × yoonittukalude gunana grouppinte gadanayekkurichu oru vivaranam nalkunnu. o" × ennathu "g" × z "r" ennathinu isomorphic aanennu prathyekamaayi ithu prasthaavikkunnu, ivide "g" ennathu "o" ennathile aikyathinte ellah verukalum ulkkollunna parimithamaaya chaakrika grouppaanu, koodaathe "r" = "r" 1 + "r" 2 − 1 (ivide "r" 1 (yathaakramam, "r" 2) "k" yude yadhaartha ulcherkkalukalude ennathe (yathaakramam, konjugettu noonn-riyal ulcherkkalukalude jodikal) soochippikkunnu. mattoru vidhathil paranjaal, "o" × ennathu "r" 1 + "r" 2−1 raankinte parimithamayi srishtikkappetta abelian grouppaanu, athinte tortion "o" leeii aikyathinte verukal ulkkollunnu.
paraspara niyamangal.
legender chinnatthinte adisthaanathil, positteev otta primukalkkulla quadratic resiprosittiyude niyamam parayunnu
quadratic resiprosity niyamatthinte saamaanyavalkkaranamaanu resiprosity niyamam.
parasparaviruddha niyamangal prakadippikkaan niravadhi vyathyastha maargangalund. pathonpathaam noottaandil kandethiya aadyakaala parasparaviruddha niyamangal saadhaaranayaayi oru power residue chinnatthinte ("p"/"q") adisthaanathil prakadippikkappettu, ithu quadratic resiprosity chinnathe saamaanyavalkkarikkunnu, ithu oru prime nambar mattoru prime aayirikkumbol vivarikkunnu. hilbert chinnangalude ("a", "b"/"p") "p" nuu mukalilulla oru uthppannam, aikyathinte verukalil moolyangal edukkunnathu, 1 nuu thulyamaanennu hilbert resiprosity niyamangal parishkarichu. aarttinte reformulated resiprosity niyamam prasthaavikkunnath aadarshangalil (allengil idels) ninnu oru galois grouppinte gadakangalilekku artin chihnam oru prathyeka upavibhaagathil nissaaramaanennu. grouppukalude cohomology allengil adelic grouppukaludeyo beejaganita keey-grouppukaludeyo praathinidyangal upayogichu niravadhi sameepakaala saamaanyavalkkaranangal paraspara niyamangal prakadippikkunnu, yadhaartha quadratic resiprosity niyamavumayulla avayude bandham kaanaan prayaasamaan.
klaas nambar formula.
klaas nambar formula oru nambar feeldinte niravadhi pradhana inavariyantukale athinte dedeckind seetta functionte oru prathyeka moolyavumaayi bandhappettirikkunnu.
bandhapetta mekhalakal.
beejaganita samkhyaasampradaayam mattu pala ganithashaasthrashaaghakalumaayumaayumaam samvadikkunnu. ithu homological beejaganithathil ninnulla upakaranangal upayogikkunnu. function feeldukalude saamyam vazhi. samkhya feeldukal, ithu beejaganita jyamithiyil ninnulla saankethikavidyakaleyum ashayangaleyum aasrayichirikkunnu. koodaathe, samkhyakalude valayangalkk pakaram z-nuu mukalilulla uyarnna alavilulla skeemukalekkurichulla padanatthe ganitha jyamiti ennu vilikkunnu. ganitha hyperbolic 3 maanifoldukalude padanatthilum beejaganita samkhya sidhaantham upayogikkunnu.

