
adisthaana samkhya sidhaantham

moolyanirnnaya-saidhaanthika reethikalkku prathyeka oonnal nalkunna beejaganita nambar sidhaanthathinteyum klaas feeld sidhaanthathinteyum oru vivaranamaya andrey weyl ezhuthiya swaadheenamulla oru pusthakamaanu adisthaana nambar sidhaantham. prinsettan sarvakalashaalayil padippicha oru kozhsinte bhagikamaaya adisthaanathil, springerude "grundlehren derr mathamattischen wissenshaften" parambarayile 144-aamm volyamaayi ithu prathyakshappettu. sameepanathiloode ellah 'eiii-feeldukalum' allengil aagola feeldukalum kaikaaryam cheiyunnu, athaayath yukthisahamaya samkhyakalude feeldinteyum oru veriyabilinte yukthisahamaya functionukalude feeldinteyum parimithamaaya beejaganita vipuleekaranangal. topological feeldukal, praadeshikamaayi othukkamulla feeldukalile haar alavinte gunangal, adelic, idelic nambar sidhaanthathinte pradhana sidhaanthangal, praadeshikavum aagolavumaaya feeldukalile lalithamaaya aaljibrakalude sidhaanthathiloode klaas feeld sidhaantham ennivayil ninnu aarambhich eee sidhaantham oru ekeekritha reethiyilaan vikasippicheduthath. thalakkettile 'basic' enna vaakk 'elementary' ennathinekkaal 'foundeshanal' enna ardhathodu aduthaanu, orupakshe vikasippicha metteeriyal automorphic roopangalude sidhaanthangal, beejaganita grouppukalude praathinidhya sidhaantham, beejaganita samkhya sidhaanthathile kooduthal noothana vishayangal ennivayude vikasanathinu adisthaanamaanenna ardhathil ithu mikacha reethiyil vyaakhyaanikkappedunnu. eee sidhaanthathinte yukthiparamayi yojicha vikasanathil oru idungiya kendreekaranathode, adisthaanaparamaayi udaaharanangalonnumillaath, eee shaili karkkasamaanu.
ganithashaasthraparamaaya sandarbhavum uddeshyavum.
aamukhathil, beejaganita samkhya sidhaanthathe hekkinte classical chikilsayil mechappeduthunnathinulla "vyarthavum asaadhyavumaaya dauthyam" ennathinupakaram, "kazhinja muppath varshathe sambhavavikasangalil ninnu nigamanagalil ethaan addeham shramichu, athil praadeshikamaayi othukkamulla grouppukal, alavu, samyojanam enniva classical nambar sidhaanthathil kooduthal pradhana punk vahikkunnu". hensell, hasse, chevalli, artin, ivasava, tettu, tamagawa ennivayude srishtikalil ninnu valarnna oru kaazhchappaad weyl vishadeekarikkunnu, athil yadhaartha samkhyakale ananthamaaya niravadhi vyathyastha poorttheekaranangalil onnaayi kaanaamenkilum vividha pii-aadik poorttheekaranangale anukoolikkaan yukthisahamaya kaaranamonnumilla. eee krameekaranatthil, adelukal (allengil moolyanirnnaya vektarukal) praadeshikamaayi oru swaabhaavika compact ring nalkunnu, athil ellah moolyanirnayangalum "oru pothu aavashyathinaayi sahakarikkunna" orotta yojicha reethiyil orumichu konduvarunnu. oru peedatthil ninnu yadhaartha samkhyakal neekkam cheyyukayum avaye pii-aadik samkhyakalkkoppam sthaapikkukayum cheiyunnathu swaabhaavikamaayum nayikkunnu-"nambar-feeldukalumaayulla poornnamaayum oresamayam chikilsayil" parimithamaaya feeldukalilulla function feeldukalude sidhaanthathinte vikasanathilekku "athu pookunnu". 1967ll amerikkayil ezhuthiya oru aamukhathinu sradheyamaaya padangal thiranjedukkunnathil, aagola mekhalakalude rund vibhagangalkkum "poornnamaayum oresamayam chikitsa nalkanam" ennu vishadeekarichukondu rachayithaav eee prathyeka kaazhchappaadine veettilekku nayikkaan thiranjedukkunnu...] verthiricha padavikku pakaram, ettavum mikachathum, ithuvare avarude bhagamaayirunna prathyekavum ennaal thulyavumaaya soukaryangal. atharam perumaattatthil tholkkunnathinu pakaram, rund vamshangalum athil nettamundaakkumennu njaan prathekshikkunnu, eee pusthakatthil ninnu vyakthamaayi uyarnnuvarum."
randaam lokamahaayudhathinushesham, klaas feeld sidhaanthathile niravadhi sambhavavikasangal klaas feeld sidhaanthathinte thelivukalil nambar feeldinte adisthaanathil nirvachikkappedunna chaakrika aaljibrakalude (kooduthal pothuve, cross cheytha uthppanna aaljibrakal) praadhaanyam kurachu. pakaram, cohomological aupachaarikatha praadeshikavum aagolavumaaya feeld sidhaanthathinte kooduthal pradhanappetta bhaagamaayi maari, prathyekichu hoach child, nakayaama, weyl, artin, tettu ennivayude pravarthanangalil 1950-1952 kaalaghattatthil.
parimithamaaya feeldukalil function feeldukalkkoppam beejaganita nambar feeldukal pariganikkanulla aagrahathodoppam, chevalliyude pravartthanatthinum prathyeka oonnal nalkunnu. praadeshika klaas feeld sidhaanthathil ninnu aagola klaas feeld sidhaanthathinte sidhaanthangal uruthirinjedukkunnathi, hasseyude nirddeshaprakaaram chevli pinneedu aidil ennu vilikkapetta element ideal avatharippichu. ananthamaaya vipuleekaranangalkkayi aagola klaas feeld sidhaanthathe vivarikkunnathinayi chevalli aanu oru nambar feeldinte aidil group aadyamaayi avatharippichathu, ennaal niravadhi varshangalkk sesham addeham praadeshika klaas feeld sidhaanthathil ninnu aagola klaas feeld sidhaantham uruthirinjedukkaan ithu oru puthiya reethiyil upayogichu. thaan upayogikkunna chila chikilsaareethikalil eee (prasidheekarikkappedaathu) kruthi kaaryamaaya swaadheenam cheluthiyathaayi weyl paraamarshichu.
sweekaranam.
onnaam pathipp ganithashaasthra avalokanangalkkayi jorjju wapples centralblaattinaayi helmut kochum avalokanam cheythu. pilkkala pathippukal fernando kyu avalokanam cheythu. ganithashaasthra association oof amerikkaykkum dablyu. centralblaattinaayi sink aandu helmut coach; randaam pathippinte avalokanathil coach eee paramarsham nadathunnu "1967 leeii sharathkaalathil moscoyil vachu shafferevich enikku aadya pathipp kaanichu, eee pusthakam inii muthal klaas feeld sidhaanthathekkurichulla pusthakathilundaakumenna paranju". chikilsayude yojip athinte chila savisheshathakalum niravadhi niroopakar eduthukaanichu, "eee pusthakam naalpathukalude thudakkathile aathmaavilaanu ezhuthiyirikkunnathu, ithu samkhya, pravartthana mekhalakalumaayi bandhapetta prashnangalekkurichu pravartthikkunna ellaavarkkum ithu oru moolyavathaaya vivara srotassaayi maattunnu."
ulladakkam.
ekadesham paranjaal, pusthakatthinte aadya pakuthi adelic, aidi "è" lick reethikalude sthiramaaya upayogathilum parimithamaaya feeldukalilulla beejaganita nambar feeldukaludeyum yukthisahamaya function feeldukaludeyum oresamayam kaikaaryam cheiyunnathilum aadunikamaanu. randaam pakuthi cohomology bhaashayillathe, prathyekichu galois cohomology bhaashayillathe lalithamaaya aljebrakaludeyum klaas feeld sidhaanthathinteyum vikasanathil aadhunikathaykk munbullathaanu. atharamoru sameepanam chittayayi vikasippikkunnathiloode eee prathyeka yaathraykk nannayi sajjeekarichirikkunna oru kappalil dhaaraalam anaavashya yanthrangal kayattumaayirunnu; kadal yaathraykk kooduthal yogyamakkunnathinupakaram athu mungiyittundaakam" ennu vishadeekarichukondu rachayithaav ithu oru idapaadaayi amgeekarikkunnu.klaas feeld sidhaanthathinte chikitsa commutative feeldukalilum lalithamaaya aaljibrakalilum vishakalana reethikal upayogikkunnu. k/k ennathu a-feeldukalude parimithamaaya saadhaarana vipuleekaranamaanenkil, k-nuu mukalilulla k-yude eathu automorphisavum k-yude ananthamaaya niravadhi sthalangalkkulla frobinius automorphisamaanu prerippikkunnathenn aadyathe ekeekritha thelivu nalkunnathil eee reethikal avarude shakti kaanikkunnu. eee sameepanam beejaganita prasthaavanakalude ganyamaayi lalithavum kooduthal yukthisahavumaya thelivu anuvadikkunnu, udaaharanathinu oru eiii-feeldinu mukalilulla oru lalithamaaya beejaganitam praadeshikamaayi ellaayidathum vibhajikkukayaanenkil maathram (aagolathalathil) vibhajikkunnu. lalithamaaya aaljibrakalude chittayaya upayogam praadeshika klaas feeld sidhaanthathinte chikitsa lalithamaakkunnu. udaaharanathinu, 2-alcohomology klaasukalude anubandha prasthavana theliyikkunnathinekkaal oru praadeshika feeldil oru lalithamaaya beejaganithathinu oru nirathirahithamaya vibhajana feeld undennu theliyikkunnathu kooduthal neraanu.
adthyaayam I.
oru parimitha vibhajana valayam commutative aanennathinte ('veddarbens littil sidhaantham') thelivu veddarbeninte roopeekaranathil ninnaan pusthakam aarambhikkunnath. local feeldukal' (oru noonn-discreet topologyil praadeshikamaayi kompaakt cheytha commutative feeldukal) eiii-feeldukalude poorttheekaranamaanennu theliyikkaan haar alavukalude gunangal upayogikkunnu. prathyekichum-pinneedu vikasippichedutha oru ashayam-avaa krithyamaayi aagola sidhaanthathinu praadeshika klaas feeld sidhaantham aavasyamulla mekhalakalaanu. praadeshikamaayi compact feeldukalallaatha noonn-commutative feeldukal oru praadeshika feeldil parimithamaaya alavilulla division aaljibrakalaanu.
adthyaayam II.
praadeshika feeldukalilulla parimitha-dimensional vekttar speysukalum topology keezhilulla division aaljibrakalum feeldinte topology nirnnayikkunna savisheshamaayi padikkukayum laattisukale topologically nirvachikkukayum cheiyunnu, minkovsky sidhaanthathinte oru analog eee pashchaathalatthil theliyikkappedunnu, koodaathe eee vekttar speysukalude caractor grouppukalekkurichulla pradhana sidhaanthangal, commutative oru dimensional kesil praadeshika feeldukalkku 'svayam duvality' aayi kurayunnu.
adthyaayam III.
oru feeldinte sthalangalude vipuleekaranangal feeldinte parimithamaaya verthirikkaavunna vipuleekarana sthalangalilekku padikkaan tensor uthppannangal upayogikkunnu, kooduthal sankeernamaaya verthirikkaanaavaatha cases pinneedilekku maattivaykkunnu.
adthyaayam IV.
eee adhyayam oru eiii-feeldinte topological adel ring, aidal group enniva avatharippikkukayum 'pradhana sidhaanthangal' inipparayunna reethiyil theliyikkukayum cheiyunnu
moolyanirnnayathinte adisthaanathil unitukale vivarikkunna eiii-feeldukalkkulla oru saamaanyavalkkaricha yoonit sidhaanthathodeyaanu adhyayam avasaanikkunnath.
adhyayam vii.
eee adhyayam nambar feeldukaludeyum function feeldukaludeyum oresamayam kaikaaryam cheiyunnathil ninnu alppam vyathyasthamaan. nambar feeld krameekaranatthil, lattisukal (athaayath fraakshanal idealukal) nirvachikkappedunnu, koodaathe oru laattisinte adisthaana domaininte haar alavu vooliyam kandethunnu. oru vipuleekaranathinte vivechanam padikkaan ithu upayogikkunnu.
adhyayam vi.
eee adhyayam function feeld kesil shradha kendreekarikkunnu; riman-roach sidhaantham alavu-saidhaanthika bhashayil prasthavikkukayum theliyikkukayum cheiyunnu, canonical klaas ulchertha feeldil nissaramaaya adile valayathinte nissaaramallaattha pratheekangalude vibhaajakangalude klaas aayi nirvachikkappedunnu.
adthyaayam VII.
oru eiii-feeldinulla seetta, ell-functionukal (samanamaaya vishakalana vasthukkal) aidil groupile integralukalude adisthaanathil prakadippikkunnu. ellah moolyanirnayangalilum eee integralukale uthppannangalayi vighadippikkukayum forrier parivartthanangal upayogikkukayum cheiyunnathu meromorphic thudarchakalkkum pravartthana samavakyangalkkum kaaranamaakunnu. udaaharanathinu, ithu muzhuvan thalatthilekkum dedeckind seetta-functionte vishakalanaparamaaya thudarchayum athinte pravartthana samavaakyavum nalkunnu. ivideyulla chikitsa aathyanthikamayi aarttinte nirddeshathilekku madangunnu, ithu tettinte prabandhatthil vikasippicheduthaanu.
adthyaayam VIII.
praadeshikavum aagolavumaaya vyathyasangalkkum vivechanagalkkumulla soothravaakyangal, ramification sidhaantham, oru function feeldinte beejaganita vipuleekaranathinte janussinulla soothravaakyam enniva vikasippicheduthittundu.
adthyaayam Ix.
chaakrika gadakangalude vyakthamaaya niyamangal ulppede lalithamaaya aaljibrakalude oru hraswa chikitsa nalkiyittundu.
x, xi ennee adhyayangal.
oru eiii-feeldil oru lalithamaaya beejaganithatthinte seetta-function nirvachikkappedunnu, koodaathe oru eiii-feeldil oru lalithamaaya beejaganithathil maximal idealukalude norm groupilum groupoidilum kooduthal phalangal theliyikkaan upayogikkunnu.
adthyaayam XII.
oru feeldinte gunana groupinteyum feeldinte beejaganita klosarinte kevala galois groupinte prateeka groupinteyum jodiyaakkalinte pashchaathalatthil oru praadeshika feeldil praadeshika klaas feeld sidhaanthathinte paraspara niyamam theliyikkappedunnu. abelian vipuleekaranangalkkaayulla ramification sidhaantham vikasippicheduthittundu.
adthyaayam XIII.
praadeshika feeldukalude gunana grouppukale a-feeldukalude aidil klaas grouppukalumaayi maattisthaapikkunna xii-aamm adhyaayathinte jody upayogichaanu a-feeldukalkkaayulla aagola klaas feeld sidhaantham vikasippicheduthath. praadeshika haash inavariyantukalude sthalangalil oru ulppannamaayaanu eee jodi nirmmichirikkunnathu.
moonnaam pathipp.
chila paraamarshangal cherkkunnu, chila cheria thiruthalukal varuthunnu, chila abhipraayangal cherkkunnu, anchu anubandhangal ulppeduthiyittundu, athil inipparayunna metteeriyal adangiyirikkunnu

