
feeled (ganitam)

ganitamlo, oka kshetram anedi oka samiti, dheeni meeda jodimpu, vyavakalanam, gunakaaram mariyu vibhajana nirvachinchabadataayi mariyu hetubaddhamaina mariyu vaastava sankhyalapai sambandhita karyakalapala vale pravartistaayi. oka kshetram anedi oka praathamika beejaganita nirmaanam, idhi beejaganitam, sankhya siddhaantam mariyu ganitamlooni aneka itara rangaalalo vistrutamgaa upayoginchabadutundi.
hetubaddha sankhyala kshetram, vaastava sankhyala kshetram mariyu sanklishta sankhyala kshetram atyanta prasiddha kshetraalu. hetubaddhamaina vidhula kshetraalu, beejaganita function kshetraalu, beejaganita sankhya kshetraalu mariyu "p"-addic kshetraalu vanti aneka itara kshetraalu saadhaaranamgaa ganitamlo, mukhyamgaa sankhya siddhaantam mariyu beejaganita jyaamitiloo upayoginchabadataayi mariyu adhyayanam cheyabadataayi. chaalaa cryptographic protocols parimita kshetraalapai aadhaarapadataayi, antey.ii., chaalaa muulakaalatoe kuudina kshetraalu.
rendu kshetraala sambandham kshetra vistarana bhavana dwara vyakteekarinchabadutundi. 1830laloe everist galois praarambhinchina galois siddhaantam, kshetra vistaranala samaruupatalanu ardham chesukovadaniki ankitam cheyabadindi. itara phalitaallo, eee siddhaantam konam trisection mariyu vruttaanni diksuchi mariyu strait edjtho cheyalemani choopistundi. antekakunda, quintic sameekaranaalu saadhaaranamgaa beejaganitaparamgaa parishkarinchalenivi anie idhi choopistundi.
aneka ganita kshetraalalo kshetraalu punaadi bhavanalaga panicheestaayi. indulo ganita vislaeshanha yokka vividha saakhalu unnaayi, ivi adanapu nirmaanamtho kuudina kshetraalapai aadharapadi untaayi. vislaeshanhaloo praathamika siddhaantaalu vaastava sankhyala kshetram yokka nirmaanaatmaka lakshanaalapai aadharapadi untaayi. mukhyamgaa beejaganita prayojanaala kosam, edainaa kshetraanni vector space kosam scalarsga upayoginchavacchu, idhi leanier beejaganitam kosam praamaanika saadhaarana sandarbham. sankhya kshetraalu, hetubaddha sankhyala kshetramloni thobuttuvulu, sankhya siddhaantamlo lothugaa adhyayanam cheyabadataayi. function feeldlu rekhaganita vastuvula lakshanaalanu vivarinchadaaniki sahaayapadataayi.
nirvachanam.
anadhikarikamga, oka kshetram anedi aaa samitilo nirvachinchabadina rendu kaaryakalaapaalato kuudina oka samitih oka jodimpu operation ilaa vraayabadindi, mariyu gunana operation ilaa vraayabadindi, rendoo hetubaddhamaina sankhyalu mariyu vaastava sankhyala kosam pravartinchetappudu, anni muulakaalaku sankalitam vilomam yokka unikitho sahaa, mariyu prati sunna kaani muulakaaniki gunakam vilomam. idhi upakalayam mariyu vibhajana yokka "viloma" karyakalapalu anie pilavabade vaatini kooda nirvachinchadam dwara pariganinchadaaniki anumatistundihi
classic nirvachanam.
adhikarikamga, oka kshetram anedi "jodimpu" mariyu "gunakaaram" anie piluvabade rendu binary kaaryakalaapaalato kuudina samiti. binary operation anedi mapping, antey, oka pratyaekamaina nirnayaatmaka moolakam yokka prati arder chesina jatha muulakaalatoe anubandhinche oka correspondence. mariyu yokka adanamgaa yokka phalitaanni mariyu yokka mottam anie pilustaaru mariyu suuchimchabadutumdi. adevidhamgaa, gunakaaram yokka phalitaanni mariyu yokka utpatti anie pilustaaru, mariyu suuchimchabadutumdi ledhaa. eee krindi lakshanaalanu santrupti parachadaniki eee karyakalapalu avasaram, veetini "kshetra suutraalu" anie pilustaaru (eee suutraalaloo, mariyu kshetramloni ekapaksha amsaalu):
dheenini ilaa sankshiptamgaa cheppavachhuh oka kshetraaniki rendu commutative operations untaayi, veetini addition mariyu multiplication anie pilustaaru; idhi aditive identityga 0 thoo adanamgaa unna samuuham; nanziro elements anedi multiplication kinda unna samuuham, gunakam identityga 1 thoo; mariyu multiplication anedi adanamgaa pampinhee avutundi.
marinta sankshiptamgaah oka kshetram anedi oka commutative ring, ikkada suutram _ 1 mariyu anni sunna kaani muulakaalu gunanamlo vilomamgaa untaayi.
pratyaamnaaya nirvachanam.
kshetraalanu kooda vibhinnamaina, conny samaanamaina margallo nirvachinchavacchu. pratyamnaayamgaa oka kshetraanni naalugu binary karyakalapala (adanamgaa, vyavakalanam, gunakaaram mariyu vibhajana) mariyu vaati avasaramaina lakshanaala dwara nirvachinchavacchu. nirvachanam prakaram, sunna dwara vibhajinchadam minahaayinchabadindi. astitva parimaanaalanu nivaarinchadaaniki, kshetraalanu rendu binary karyakalapalu (adanamgaa mariyu gunakaaram), rendu ekasambandha karyakalapalu (varusaga sankalana mariyu gunakaara vilomaalanu istundi), mariyu rendu shoonya karyakalapalu (sthiraamkaalu mariyu) dwara nirvachinchavacchu. eee karyakalapalu appudu paina perkonna paristhitulaku lobadi untaayi. nirmaanaatmaka ganitam mariyu gananalo astitva parimaanaalanu nivaarinchadam mukhyam. okati oke rendu binary operationla dwara oka feeldnu samanamga nirvachinchavacchu, oka yeka operation (gunakam vilomam), mariyu rendu (tappanisarigaa vibhinnamgaa undakudadu) sthiraamkaalu mariyu, nundi mariyu.
udaaharanalu.
hetubaddhamaina sankhyalu.
kshetra bhavana yokka vistaranaku chaalaa kaalam mundu hetubaddhamaina sankhyalu vistrutamgaa upayoginchabaddaayi.
avi bhinnaluga vrayagala sankhyalu
, ekkada mariyu puurnaamkaalu, mariyu. atuvanti bhinnam yokka sankalana vilomam, mariyu gunakaara vilomam (adhi andinchina), idhi eee krindi vidhamgaa chudavachuhu
nairuupyamgaa avasaramaina kshetra suutraalu hetubaddha sankhyala praamaanika lakshanaalaku taggutaayi. udaaharanaku, pampinhee yokka chattanni eee krindi vidhamgaa niroopinchavacchuh
vaastava mariyu sanklishta sankhyalu.
jodimpu mariyu gunakaaram yokka saadhaarana kaaryakalaapaalato vaastava sankhyalu kooda oka kshetraanni yerparustaayi. sanklishta sankhyalu vyakteekaranalanu kaligi untaayi
oohaatmaka unit ekkada undhi, i.ii., santruptikaramaina (vaastavamaina) sankhya.
vaastava sankhyala yokka jodimpu mariyu gunakaaram eee rakamaina vyakteekaranalu anni kshetra suutraalanu santrupti parache vidhamgaa nirvachinchabaddaayi mariyu tadwara avi pattukuntaayi. udaaharanaku, pampinhee chattam amalu chestundi
idhi mallii paina perkonna rakam yokka vyakteekarana anie takshaname cheppavacchu, kabatti sanklishta sankhyalu oka kshetraanni yerparustaayi. sanklishta sankhyalanu rekhaaganitaparamgaa samatalamlo binduvulugaa suuchimchavachchu, cartician akshaamsaalu vaati vyakteekarana yokka vaastava sankhyala dwara ivvabadataayi, ledhaa muulam nundi eee binduvula varaku banaluga, vaati podavu mariyu kontha pratyaekamaina disatoe chuttabadina konam dwara paerkonabadataayi. adanamgaa baanaalanu sahajamaina samaantara chaturbhujaaniki kalapadam (cartician coordinatelanu jodinchadam), mariyu gunakaaram-takkuva sahajamgaa-baanaala bhramana mariyu scalingnu kalapadam (koonaalanu jodinchadam mariyu podavulanu guninchadam). vaastavika mariyu sanklishta sankhyala rangaalu ganitam, bhouthika shaastram, injaneering, ganaamkaalu mariyu aneka itara shaastreeya vibhagalalo upayoginchabadataayi.
nirmaanaatmaka sankhyalu.
puraatana kaalamlo, diksuchi mariyu strait edjtho nirdishta sankhyalanu nirminche saadhyaasaadhyaalaku sambandhinchina aneka rekhaganita samasyalu unnaayi. udaaharanaku, icchina koonaanni eee vidhamgaa tribhujaakaaramlo vibhajinchadam saadhaaranamgaa asaadhyamani greekulaku teliyadu. eee samasyalanu nirmaanaatmaka sankhyala kshetraanni upayoginchi parishkarinchavacchu. nijamaina nirmaanaatmaka sankhyalu, nirvachanam prakaram, diksuchi mariyu strait edjnu matrame upayoginchi 0 mariyu 1 paayintla nundi aneka dasalalo nirminchagala line segments yokka podavulu. eee sankhyalu, vaastava sankhyala kshetra kaaryakalaapaalato sampannamainavi, nirmaanaatmaka sankhyalaku parimitam cheyabaddaayi, oka kshetraanni yerparustaayi, indulo hetubaddha sankhyala kshetram sarigga untundi. eee drushtaantam nirmaanaatmaka sankhyala varga muulaala nirmaanaanni chooputundi, idhi tappanisarigaa lopala undakudadu. drushtaamtamlo lebulnu upayoginchi, vibhaagaalanu nirminchandi, mariyu oka ardha vruttam (madhya binduvu vadda madhyalo), idhi oka binduvu dwara lamba rekhanu chhedistundi, oka podavu unnappudu nundi sarigga suutram _ 4 dhooramlo untundi.
anni vaastava sankhyalu nirmaanaatmakamainavi kaavu. suutram _ 5 anedi nirmaanaatmaka sankhya kaadani chuupavachchu, idhi diksuchi thoo nirminchadam mariyu ghanapu vaipu yokka podavunu valume 2 thoo neerugaa cheyadam asadhyam anie suuchistumdi, idhi puraatana greekulu edurkonna maroka samasya.
naalugu muulakaalatoe kuudina kshetram.
hetubaddhata vanti suparichitamaina sankhya vyavasthalatho paatu, kshetraalaku itara, takkuva takshana udaaharanalu unnaayi. eee krindi udaaharana,,,, mariyu anie piluvabade naalugu muulakaalatoe kuudina kshetram. sangnaamaanam anedi sankalana gurtimpu moolakam (paina perkonna suutraalaloo 0 anie suuchimchabadutumdi) paatranu poeshinchae vidhamgaa enchukobadutundi, mariyu gunakaara gurtimpu (paina perkonna suutraalaloo 1 anie suuchimchabadutumdi). kshetra suutraalanu marikonni kshetra siddhaantaanni upayoginchadam dwara ledhaa pratyaksha ganana dwara dhruveekarinchavachu. udaaharanaku,
eee kshetraanni naalugu muulakaalatoe kuudina parimita kshetram anie pilustaaru, mariyu dheenini suuchistaaru ledhaa. mariyu (kudivaipuna unna pattikallo erupu rangulo highlight cheyabadindi) kaligi unna upasamiti kooda oka kshetram, dheenini "binary feeled" ledhaa anie pilustaaru. computer sains mariyu booliyan beejaganitam sandarbhamlo, mariyu tarachugaa varusaga "falls" mariyu "true" dwara suuchimchabadataayi, mariyu adanamgaa xor (exclusive ledhaa) anie suuchimchabadutumdi. maroe maatalo cheppalante, binary feeled yokka nirmaanam anedi bitlatho computingnu anumatinche praathamika nirmaanam.
praathamika bhavalu.
eee vibhaagamlo, ekapaksha kshetraanni suuchistumdi mariyu avi ekapaksha amsaalu.
nirvachanam yokka parinaamaalu.
okati mariyu undhi. mukhyamgaa, prati moolakam yokka sankalana vilomaanni telisina ventane evarainaa oohinchavachhu.
appudu ledhaa tappanisarigaa 0 ayithe, endukante, unte, appudu
. dheeni ardham prati feeled oka samagra domine.
adanamgaa, eee krindi lakshanaalu edainaa muulakaalaku vartistaayi mariyu
oka kshetram yokka sankalitam mariyu gunakam samuuham.
oka kshetram yokka suutraalu idhi adanamgaa unna abelian samuuham anie suuchistaayi. eee samuuhaanni kshetramloni sankalana samuuham anie pilustaaru, mariyu konnisaarlu daanini gandaragolamgaa undagaladani suchinchetappudu suuchistaaru.
adevidhamgaa, gunakaaram kinda abelian samuuhaanni yerparuchukune "sunna kaani" muulakaalu, gunakaara samuuham anie piluvabadataayi mariyu kevalam ledhaa dwara suuchimchabadataayi.
oka kshetraanni eee vidhamgaa jodimpugaa suchinchabade rendu kaaryakalaapaalato kuudina samitigaa nirvachinchavacchu mariyu adanamgaa unna abelian samuuham vanti gunakaaram, gunakaaram kinda unna abelian samuuham (ikkada 0 anedi jodimpu yokka gurtimpu moolakam), mariyu gunakaaram anedi adanamgaa pampinhee cheyabadutundi. anduvalla samuuhaala saadhaarana vaastavaalanu vartimpajeyadam dwara kshetraala gurinchi konni praathamika prakatanalanu pondavacchu. udaaharanaku, sankalana mariyu gunakaara vilomaalu mariyu pratyekinchi dheeni dwara nirnayinchabadataayi.
avasaram anusaristundi, endukante 1 anedi 0 laeni samuuham yokka gurtimpu moolakam. anduvalla, oke muulakamtoe kuudina chinna ring, oka kshetram kaadu.
oka kshetram yokka gunaka samuuham yokka prati parimita upa samuuham chakreeyamgaa untundi (chudandi).
lakshanam.
f" yokka rendu muulakaala gunanamtho paatu, oka ekapaksha moolakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka gunakam yokka
saanukuula puurnaamkam lekapothe
appudu 0 anae lakshanam undani chebutaaru. udaaharanaku, hetubaddha sankhyala kshetram 0 anae lakshanaanni kaligi untundi, endukante yea saanukuula puurnaamkam sunna kaadu. lekapothe, eee sameekaranaanni santruptipariche "saanukuula puurnaamkam" unte, atuvanti atichinna saanukuula puurnaamkaanni pradhana sankhyagaa chuupavachchu. dheenini saadhaaranamgaa suuchistaaru mariyu appudu kshetraaniki lakshanam undani chebutaaru.
udaaharanaku, feeled 2 lakshanaanni kaligi undhi (paina perkonna jodimpu pattika yokka sangnaamaanamlo).
okavela lakshanam unte, appudu andariki. idhi suuchistumdi
dwaipaakshika suutramloe kanipinche anni itara dvipada gunakaalu dheeni dwara vibhajinchabadataayi. ikkada, (karakalu) anedi-va shakti, i.ii.moolakam yokka-rettimpu utpatti. anduvalla, frobenius mapp
idhi adanamgaa (mariyu gunanamtho kooda) anukuulamgaa untundi, anduvalla idhi oka kshetra samaruupata. eee homomorphism uniki lakshanaala 0 yokka kshetraala nundi chaalaa bhinnamgaa untundi.
upakshetraalu mariyu pradhana kshetraalu.
oka feeled yokka "subfield" anedi feeled kaaryakalaapaalaku sambandhinchi oka feeled yokka upasamiti. samanamga, idhi oka upasamiti, mariyu idhi sunna kaani moolakam yokka adanamgaa, gunakaaram, sankalana vilomam mariyu gunakam vilomam kinda moosiveyabadutundi. dheeni ardham, andarikee mariyu andariki, rendintikee mariyu andariki, rendintikee mariyu rendintikee.
kshetra homomorphisms anevi rendu kshetraala madhya maplu, avi, mariyu, ekkada mariyu ekkada anevi ekapaksha amsaalu. anni kshetra homomorphisms suudi mandulu. okavela surjective ayithe, daanini isomorphism (ledhaa feeldlu mariyu isomorphic anie pilustaaru) anie pilustaaru.
oka kshetraaniki sarainadi lekapothe daanini pradhana kshetram antaaru (i.ii., khachitamgaa chinna) upakshetraalu. edainaa feeled oka pradhana feeldnu kaligi untundi. (oka pradhana sankhya) yokka lakshanam unte, pradhana kshetram krinda pravesapettina parimita kshetraaniki isomorphicga untundi. lekapothe pradhana kshetram isomorphic gaa untundi.
parimita kshetraalu.
"parimita kshetraalu" ("galois kshetraalu" anie kooda pilustaaru) anevi chaalaa muulakaalatoe kuudina kshetraalu, veeti sankhyanu kshetra kramam anie kooda pilustaaru. paina perkonna parichaya udaaharana naalugu muulakaalatoe kuudina kshetram. daani upa kshetram athi chinna kshetram, endukante nirvachanam prakaram oka kshetram kaneesam rendu vibhina muulakaalanu kaligi untundi.
pradhana kramamlo saralamaina parimita kshetraalu, modular ankaganitam upayoginchi chaalaa neerugaa andubaatulo untaayi. sthira dhanaatmaka puurnaamkam kosam, ankaganitam "maadyulo" antey sankhyalatho panicheyadam
eee samitipai jodimpu mariyu gunakaaram puurnaamkaala samitilo prasnalooni operation cheyadam dwara cheyabadutundi, phalitamgaa migilina vaatitho vibhajinchi, teesukuntaaru. eee nirmaanam oka pradhana sankhya ayithe khachitamgaa oka kshetraanni istundi. udaaharanaku, paina perkonna rangamlo pradhana phalitaalanu teesukovadam. edainaa mishrama sankhya kosam (i.ii.rendu khachitamgaa chinna sahaja sankhyala gunakam vale vyakteekarinchagala yea sankhya ayina, oka kshetram kaaduh rendu sunna kaani muulakaala gunakam sunna, endukante paina vivarinchina vidhamgaa, idhi oka kshetram kakunda nirodhistundi. eee vidhamgaa nirminchina muulakaalatoe (pradhaanamaina) kshetram saadhaaranamgaa dheeni dwara suuchimchabadutumdi.
prati parimita kshetraaniki muulakaalu untaayi, ikkada pradhana mariyu. eee prakatana daani pradhana kshetrampai vector spacega chudavachu. eee vector space yokka parimaanam tappanisarigaa parimitamainadi, cheppandi, idhi nokkicheppabadina prakatnanu suuchistumdi.
muulakaalatoe kuudina kshetraanni bahupadi yokka vibhajana kshetramgaa nirminchavacchu.
atuvanti vibhajana kshetram anedi bahupadi sunnalanu kaligi unna podigimpu. antey degree nundi veelainanni ekkuva sunnalu unnaayi. endukante, paina perkonna gunana pattikanu upayoginchi, sameekaranamloni naalugu muulakaalu santruptikaramgaa unnayani, kabatti avi sunnaluga unnayani, prati sandarbham tanikhi cheyavachu. dheeniki viruddhamgaa, rendu sunnalu matrame unnaayi (avi 0 mariyu 1), kabatti eee chinna rangamlo sarala karakaluga vibhajinchabadadu. praathamika kshetra-siddhaanta bhaavanalanu marinta vivaristuu, oke kramamlo unna rendu parimita kshetraalu isomorphic anie chuupavachchu. anduvalla muulakaalatoe kuudina parimita kshetram gurinchi matladatam aacharam, dheenini suuchistaaru ledhaa.
charitra.
chaaritraatmakamgaa, moodu beejaganita vibhagalu oka kshetram yokka bhavanaku daariteesaaih bahupadi sameekaranaalu, beejaganita sankhya siddhaantam mariyu beejaganita jyaamitini parishkarinche prasna. oka kshetram yokka bhavana vaipu modati adugu 1770loe joseph-luis lagrange chae veyabadindi, athanu vyakteekaranalo cubic polynamial yokka sunnalanu kramabaddheekarinchadaanni gamaninchaadu.
(aikyata yokka moodava muulam kaavadamtho) rendu viluvalanu matrame istundi. eee vidhamgaa, lagrange sipion dell ferro mariyu francois whit yokka shaastreeya parishkara paddhatini sambhaavitamgaa vivarinchindi, idhi teliyani daaniki cubic sameekaranaanni quadratic sameekaranaaniki tagginchadam dwara konasaagutundi. degree 4 sameekaranaala kosam idhe vidhamaina pariseelanatoo, lagrange chivariki kshetraala bhavana mariyu samuuhaala bhavanaga maarindi. wondermonde, 1770loe kooda, mariyu puurti sthaayilo, carle friedrich gaas, thana "dequisitions arthematy" (1801) loe, sameekaranaanni adhyayanam chesaaru.
oka pradhana kosam mariyu, mallii aadhunika bhashanu upayoginchi, phalitamgaa vachhe chakriya galois samuuham. okavela unte oka saadhaarana-gonnu nirminchavacchani gaas oohinchaaru. lagrange yokka panini aadhaaramgaa chesukuni, pavolo rufini (1799) quintic sameekaranaalanu (degree 5 yokka bahupadi sameekaranaalu) beejaganitaparamgaa parishkarinchalemani paerkonnaadu; ayithe, athani vaadanalu lopabhooyishtamgaa unnaayi. eee antaraalanu 1824loe neels henrique abel dwara nimpaaru. 1832loe everist galois, beejaganitaparamgaa parishkarinchadagina bahupadi sameekaranaaniki avasaramaina mariyu taginanta pramaanaalanu roopondinchaaru, tadwara nedu galois siddhaantam anie piluvabade daanini amaluloki tecchaaru. abel mariyu galois rendoo nedu beejaganita sankhya kshetram anie piluvabade daanitoe panicheesaayi, conny oka kshetram yokka spashtamaina bhavananu ledhaa samuuham yokka spashtamaina bhavananu kalpinchaledu.
1871loe richard dedeckind naalugu ankaganita karyakalapala kinda moosiveyabadina nijamaina ledhaa sanklishta sankhyala samiti kosam, german padam "corper", antey "sareeram" ledhaa "corpus" (sendreeyamgaa moosiveyabadina astitvaanni suuchimchadaaniki). aangla padam "feeled" nuu parichayam chesaaru.
1881loe leopold kronekar "hetubaddhata yokka domine" anie piliche daanini nirvachinchaadu, idhi aadhunika paramgaa hetubaddha bhinnala kshetram. kronekar yokka bhavana anni beejaganita sankhyala kshetraanni cover cheyaledu (idhi dedecind yokka ardhamlo oka kshetram), conny marovaipu idhi dedecind kante ekkuva nairuupyamgaa undhi, idhi oka kshetram yokka muulakaala swabhaavampai nirdishta oohanu cheyaledu. kronekar hetubaddhamaina function feeled vanti oka feeldnu nairuupyamgaa ardham chesukunnadu. dheeniki mundu, 1844loe joseph leoville chesina rachanala nundi, charless hermite (1873) mariyu ferdinand vaan lindeman (1882) varusaga mariyu yokka utkrushtatanu niroopinche varaku, divya sankhyala udaaharanalu telisinavi.
nairoopya kshetram yokka modati spashtamaina nirvachanam dheeniki kaaranam. mukhyamgaa, heinrich martin webber yokka bhaavanalo feeled kooda undhi. giusape veronese (1891) adhikaarika shakti shrenini adhyayanam chesaaru, idhi "p"-addic sankhyala kshetraanni pravesapettadaaniki daariteesindi. ippativaraku perukupoyina nairoopya kshetra siddhaantam yokka gnaanaanni samslaeshanha chesaaru. athanu kshetraala lakshanaalanu svayamsiddhamgaa adhyayanam chesi, aneka mukhyamaina kshetra-siddhaanta bhaavanalanu nirvachinchaadu. galois siddhaantam vibhagalalo perkonna siddhaantaalalo ekkuva bhaagam, nirmaana kshetraalu mariyu praathamika bhaavanalanu stinitz rachanalo chudavachu. oka kshetramlo kramabaddheekaranala bhavananu, tadwara vislaeshanha yokka praantaanni puurtigaa beejaganita lakshanaalatoo anusandhanincharu. emil artin 1928 nundi 1942 varaku galois siddhaantaanni tirigi abhivruddhi chesi, aadima muulaka siddhaantampai aadharapadataanni tolaginchaadu.
kshetraalanu nirminchadam.
valayala nundi kshetraalanu nirminchadam.
oka commutative ring anedi oka samiti, idhi oka jodimpu mariyu gunakaaram operation kaligi untundi, gunakam vilomaala uniki minaha, oka kshetram yokka anni suutraalanu santruptiparustundi. udaaharanaku, puurnaamkaalu oka commutative ringnu yerparustaayi, conny oka feeled kaaduh oka puurnaamkam yokka paraspara anedi oka puurnaamkam kaadu, tappa.
beejaganita nirmaanaala shrenilo kshetraalanu commutative ringulugaa vargeekarinchavachhu, dheenilo prati nanziro moolakam oka unit (antey prati moolakam invertible). adevidhamgaa, kshetraalu khachitamgaa rendu vibhina aadarsaalato commutative valayalu, mariyu. kshetraalu kooda khachitamgaa commutative valayalu, veetilo ekaika pradhana aadarsam.
commutative ring ichinatlayithe, daaniki sambandhinchina feeldnu nirminchadaaniki rendu margalu unnaayi, i.ii.anni sunna kaani muulakaalu vilomamgaa maare vidhamgaa savarinchadaaniki rendu margalu unnaih bhinnala kshetraanni yerparachadam mariyu avashesha kshetraalanu yerparachadam. bhinnala kshetram, hetubaddhamainavi, ayithe avashesha kshetraalu parimita kshetraalu.
bhinnala kshetram.
oka samagra domine ichinatlayithe, daani bhinnala kshetram, q anedi puurnaamkaala nundi nirminchinatle rendu muulakaala bhinnaalatoe nirminchabadutundi. marinta khachitamgaa, muulakaalu ekkada mariyu ekkada unnaayi, mariyu anevi bhinnalu. rendu bhinnalu mariyu unte mariyu unte matrame samanamga untaayi. bhinnaalapai operation anedi hetubaddha sankhyala maadirigaane panichestundi. udaaharanaku,
ring oka samagra domine ayithe, bhinnala samiti oka feeldnu yerparustundani choopinchadam suutigaa untundi.
oka kshetrampai (ledhaa oka samagra domine) hetubaddha bhinnala kshetram anedi bahupadi valayamlooni bhinnala kshetram. feeled off laurent syries
oka kshetrampai anedi adhikaarika shakti shreni yokka valayamlooni bhinnala kshetram (dheenilo). edainaa laurent syries anedi power syries yokka oka bhinnam, idhi power off power dwara vibhajinchabadindi (ekapaksha power siriisku viruddhamgaa), ayithe eee paristhitiloo bhinnala praatinidhyam takkuva mukhyamainadi.
avashesha kshetraalu.
bhinnala kshetraaniki adanamgaa, idhi oka kshetramlo injectivga ponduparachabadutundi, oka kshetraanni oka surjective mapp dwara oka commutative ring nundi pondavacchu. eee vidhamgaa pondina edainaa kshetram oka bhagaswamyamu, ikkada garishta aadarsam. oke oka garishta aadarsam unte, eee kshetraanni avashesha kshetram antaaru.
bahupadi valayamlo (oka kshetrampai) oke bahupadi dwara utpatti cheyabadina aadarsam, i loe irreducible ayithe matrame garishtamgaa untundi.ii., rendu bahupadaalu chinna sthaayilo utpattigaa vyakteekarinchalekapothe. idhi oka polaanni istundi
eee kshetram sameekaranaanni santruptipariche oka muulakaanni (anagaa avashesha taragati) kaligi untundi.
udaaharanaku, pakkane unna oohaatmaka unit chihnam nundi pondabadutundi, idhi ekkada santrupti chendutundi. antekakunda, idhi irreducible ovar, idhi oka isomorphismnu ivvadaaniki bahupadaanni pampe myapnu suuchistumdi.
pedda polamlo polaalanu nirminchadam.
icchina pedda container feeled lopala polaalanu nirminchavacchu. oka feeled icchinatlu anukundam, mariyu oka subfieldga unna feeled. edainaa moolakam kosam, kaligi unna atichinna upa kshetram undhi mariyu dheenini "f" yokka upa kshetram anie pilustaaru, idhi utpatti cheyabadutundi mariyu suuchimchabadutumdi. nundi vellae marganni "oka muulakaaniki prakkane" anie suuchistaaru. marinta saadhaaranamgaa, oka upasamiti kosam, kaligi unna mariyu suuchimchina kaneesa upa kshetram untundi.
rendu upakshetraala misramam mariyu konni kshetraala misramam rendintinee kaligi unna atichinna upakshetramgaa untundi mariyu misramaanni oka nirdishta aastini santruptipariche atipedda upakshetraanni nirminchadaaniki upayoginchavacchu, udaaharanaku, diguva pravesapettina bhaashalo, beejaganita uparitalam.
feeled podigimpulu.
upa kshetram yokka bhavananu vyatiraeka drukkonam nundi kooda pariganinchavacchu, dheenini soochinche "feeled podigimpu" (ledhaa kevalam podigimpu) gaa soochinchadam dwara,
mariyu "ovar" chadavandi.
feeled extension yokka praathamika daytum daani degree, i.ii., oka-vector space gaa parimaanam. idhi suutraanni santruptiparustundi
parimitamaina degree unna podigimpulanu parimita podigimpulu anie pilustaaru. podigimpulu mariyu degree 2, ayithe anantamaina podigimpu.
beejaganita podigimpulu.
kshetra vistaranala adhyayanamlo oka keelaka bhavana beejaganita amsaalu. oka moolakam anedi bahupadi yokka muulamgaa unte, adhi bahupadi sameekaranaanni santruptiparichinatlayite, adhi "beejaganita" avutundi.
thoo, mariyu.
udaaharanaku, oohaatmaka unit beejaganitaparamgaa untundi, mariyu paiga kooda, endukante idhi sameekaranaanni santruptiparustundi.
prati moolakam beejaganitamgaa unde kshetra vistarananu beejaganita podigimpu antaaru. paina perkonna gunakaara suutram nundi oohinchagaliginatlugaa, edainaa parimita podigimpu tappanisarigaa beejaganitamgaa untundi.
paina perkonna vidhamgaa, oka moolakam dwara utpatti cheyabadina upa kshetram, oka beejaganita moolakam ayithe mariyu unte matrame beejaganita podigimpu. antey, beejaganitamaite, anni itara muulakaalu kooda tappanisarigaa beejaganitame. antekakunda, podigimpu yokka degree, i.ii., oka-vector space yokka parimaanam, paina perkonna vidhamgaa, oka bahupadi sameekaranam unna kaneesa degreeki samanam. eee degree unte, appudu muulakaalu roopam kaligi untaayi
udaaharanaku, gassian hetubaddhata yokka kshetram anedi rendu hetubaddhamaina sankhyalu unna roopam yokka anni sankhyalanu kaligi unna upa kshetram roopam yokka saaraamsaalanu (mariyu adevidhamgaa adhika ghaataankaalaku) ikkada pariganinchaalsina avasaram ledhu, endukante dheenini saralikrutam cheyavachu.
atindriya sthaavaraalu.
paina perkonna hetubaddha bhinnala kshetram, ikkada oka anischitamainadi, idhi beejaganita podigimpu kaadu, endukante sunnaalo gunakaalato bahupadi sameekaranam ledhu. beejaganitam kaani muulakaalanu divyamainavi antaaru. anadhikarikamga cheppalante, anischita mariyu daani adhikaralu daani amsaalatoe sankarshana chendavu. ilanti nirmaanaanni kevalam okadaaniki badulugaa anischitamaina nirmaanaalatoo nirvahinchavacchu.
marosari, paina charchinchina feeled podigimpu oka mukhya udaaharanah beejaganitam kakapothe (i.ii., anedi gunakaalato kuudina bahupadi yokka muulam kaadu), appudu dheeniki isomorphic. eee isomorphism hetubaddha bhinnalaku pratyamnaayamgaa pondabadutundi.
oka kshetram yokka upasamiti beejaganitaparamgaa swatantramgaa unte (yea bahupadi sambandhaalanu santrupti parachakapothe) mariyu beejaganita podigimpu ayithe adhi oka atindriya aadharam. edainaa kshetra vistaranaku oka atindriya aadharam untundi. anduvalla, kshetra podigimpulanu (puurtigaa divya podigimpulu) mariyu beejaganita podigimpulugaa vibhajinchavachhu.
moosiveta karyakalapalu.
oka kshetram khachitamgaa pedda beejaganita podigimpulu lekunte beejaganitaparamgaa moosiveyabadutundi ledhaa, samanamga, edainaa bahupadi sameekaranam unte
oka parishkaram undhi. beejaganitam yokka praathamika siddhaantam prakaram, beejaganitaparamgaa moosiveyabadindi, i.ii.sanklishta gunakaalato edainaa bahupadi sameekaranaaniki sanklishta parishkaram untundi. hetubaddhamaina mariyu vaastava sankhyalu sameekaranam nundi beejaganitaparamgaa "moosiveyabadavu
dheeniki etuvanti hetubaddhamaina ledhaa nijamaina parishkaram ledhu. idhi beejaganitamgaa unte (sthuulamgaa cheppalante, daanitoe poliste chaalaa peddadi kaadu) mariyu beejaganitaparamgaa moosi unte (anni bahupadi sameekaranaala parishkaralanu kaligi undentha peddadi) daanini "beejaganita moosiveta" anie pilustaaru.
paina perkonna vidhamgaa, idhi beejaganita moosiveta. beejaganita moosiveta anedi kshetram yokka parimita podigimpu anae paristhiti chaalaa pratyekamainadih artin-scrayer siddhaantam prakaram, eee podigimpu yokka degree tappanisarigaa 2, mariyu muulakamgaa dheeniki samanam. atuvanti kshetraalanu real closed kshetraalu anie kooda pilustaaru.
edainaa kshetraaniki beejaganita moosiveta untundi, idhi (pratyekamainadi kaanidi) isomorphism varaku pratyaekamgaa untundi. dheenini saadhaaranamgaa "beejaganita moosiveta" gaa suuchistaaru mariyu suuchistaaru. udaaharanaku, beejaganita moosivetanu beejaganita sankhyala kshetram antaaru. daani nirmaanaaniki ultrafilter lemma, empika yokka suutram kante balaheenamaina sett-theoretic suutram avasaram kabatti kshetram saadhaaranamgaa avyaktamgaa untundi. eee vishayamlo, beejaganita moosiveta anoohyamgaa sulabham. idhi (kramam yokkavi) kaligi unna parimita kshetraala union. 0 anae lakshanam yokka edainaa beejaganitamgaa moosivesina kshetraaniki, laurent shreni yokka beejaganita moosiveta anedi puisex shreni yokka kshetram, idhi daani prakkane unna muulaala dwara pondabadutundi.
adanapu nirmaanamtho kuudina kshetraalu.
ganitamlo mariyu antaku minchi kshetraalu sarvavyaaptam kabatti, eee bhavana yokka aneka merugudalalu nirdishta ganita praantaala avasaralaku anugunamgaa marai.
arder chesina kshetraalu.
edainaa rendu muulakaalanu polchagaligithe, oka feeled "f" nuu "arder feeled" anie pilustaaru, tadwara mariyu eppudu mariyu eppudu. udaaharanaku, vaastava sankhyalu saadhaarana kramamtho kramabaddheekarinchina kshetraanni yerparustaayi. artin-shriyar siddhaantam oka kshetraanni adhi adhikarikamga nijamaina kshetramaite matrame aadesinchavacchani perkondi, antey edainaa chaturasra sameekaranam anie ardham.
dheeniki parishkaram matrame undhi. sthira kshetrampai saadhyamayye anni aadesaala samiti, chaturasraakaara roopaala yokka whit ring nundi ring homomorphisms samitiki isomorphicga untundi.
archimedian feeled anedi prati muulakaaniki parimita vyakteekarana unde kramabaddhamaina feeled.
dheeni viluva aaa moolakam kante ekkuvagaa untundi, antey, anantamaina muulakaalu laevu. samanamga, eee kshetramlo anantamainavi (anni hetubaddha sankhyala kante chinna muulakaalu) laevu; ledhaa, inca samanamga, eee kshetram oka upakshetraaniki samaruupamgaa untundi.
anni eguva sarihaddulu, diguva sarihaddulu (taggimpu cutt chudandi) mariyu parimitulu, unikilo unte, oka kramabaddheekarinchina kshetram taggimpu-puurti avutundi. marinta adhikarikamga, prati sarihaddu upasamiti kaneesam eguva sarihaddunu kaligi undaali. edainaa puurti kshetram tappanisarigaa archimedian ayi undaali, endukante edainaa naan-archimedian feeldlo goppa anantamainadi ledhaa kaneesam saanukuula hetubaddhamainadi undadu, ikkada nundi prati moolakam prati anantam kante ekkuvagaa unna kramamlo, etuvanti parimiti undadu.
reals yokka prati saraina upa kshetram kooda atuvanti antaraalanu kaligi unnanduna, isomorphism varaku pratyaekamaina puurti kramabaddheekarinchina kshetram. calculuslo aneka punaadi phalitaalu vaastavaalu yokka eee lakshanaala nundi neerugaa anusaristaayi.
hyperrials archimedian kaani arder feeldnu yerparustaayi. idhi anantamaina mariyu anantamaina sankhyalanu cherchadam dwara pondina vaastavaala podigimpu. ivi yea vaastava sankhya kante varusaga chinnaviga untaayi. hyperrials praamaanikam kaani vislaeshanha yokka punaadi aadharaanni yerparustaayi.
sthalashaastra rangaalu.
oka kshetram yokka bhavana yokka maroka merugudala oka sthalavignaana kshetram, dheenilo samiti oka sthalavignaana sthalam, antey kshetram yokka anni karyakalapalu (adanamgaa, gunakaaram, pataalu mariyu) sthalam yokka sthalavignaanaaniki sambandhinchi nirantara pataalu.
krinda charchinchina anni rangaala topology oka metrik nundi prerepinchabadindi, i.ii.oka function
idhi edainaa rendu muulakaala madhya "dhooram" nuu kolustundi.
puurti cheyadam anedi maroka rangam, indulo, anadhikarikamga cheppalante, asalu kshetramloni "khaaliilu" evaina unte, nimpabadataayi. udaaharanaku, edainaa ahetuka sankhya, antey, hetubaddhamaina sankhyalalo oka "gyap", idhi oka vaastava sankhya, idhi hetubaddhamaina sankhyala dwara sumaruga anchanaa veyavachu, antey sampoorna viluva dwara ivvabadina dhooram kaavalasinanta chinnadi.
eee krindi pattika eee nirmaanaaniki konni udaaharanalanu jabita chestundi. nalgava kaalam sunna kramam yokka udaaharananu chooputundi, i.ii., dheeni parimiti (kosam) sunna ayina kramam.
eee kshetram sankhya siddhaantam mariyu-addic vislaeshanhaloo upayoginchabadutundi. beejaganita moosiveta oka pratyaekamaina pramaanaanni kaligi untundi, conny adhi puurti kaadu. ayithe, eee beejaganita moosiveta puurti ayipovadam anedi beejaganitaparamgaa moosiveyabadutundi. sanklishta sankhyalaku daani sthuula saaruupyata kaaranamgaa, dheenini konnisaarlu sanklishta p-adika sankhyala kshetram anie pilustaaru mariyu dheeni dwara suuchistaaru.
sthaanika kshetraalu.
eee krindi sthalavignaana kshetraalanu "sthaanika kshetraalu" anie pilustaaruh
eee rendu rakaala sthaanika rangaalu konni praathamika saaruupyatalanu panchukuntaayi. eee vishayamlo, muulakaalu mariyu (uniformiser anie suuchistaaru) okadaanikokati anugunamgaa untaayi. dheeni modati abhivyakti praathamika sthaayilo untundih rendu kshetraala muulakaalanu uniformiserlo power syrisga, gunakaalato vyakteekarinchavachhu. (ayithe, jodimpu mosukelladam dwara cheyabadutundi, idhi kesu kaadu, eee kshetraalu isomorphic kaadu.) eee krindi vaastavaalu eee uparitala saaruupyata chaalaa lothugaa velutundani chuputai
differential fields.
avakalana kshetraalu anevi oka utpannam kaligina kshetraalu, i.ii., kshetramloni muulakaala utpannaalanu teesukoovadaaniki anumatinchandi. udaaharanaku, polynomials yokka praamaanika utpannamtho kalisi feeled oka avakalana kshetraanni yerparustundi. eee kshetraalu avakalana galois siddhaantaaniki kendramgaa unnaayi, idhi sarala avakalana sameekaranaalato vyavaharinchae galois siddhaantam yokka oka roopaantaram.
galois siddhaantam.
galois siddhaantam anedi jodimpu mariyu gunakaaram yokka ankaganita karyakalapalalo samaruupatanu adhyayanam cheyadam dwara oka kshetram yokka beejaganita podigimpulanu adhyayanam chestundi. eee praantamloo oka mukhyamaina bhavana emitante, parimita galois podigimpulu, ivi nirvachanam prakaram, vaeru cheyadaginavi mariyu saadhaaranamainavi. aadima muulaka siddhaantam parimita vibhajana podigimpulu tappanisarigaa saralamainavani choopistundi, i.ii., roopam
ikkada irreducible polynamial (paina perkonna vidhamgaa) untundi. atuvanti podigimpu kosam, saadhaarana mariyu vaeru cheyadaginadi antey anni sunnalu kaligi untaayi mariyu saralamaina sunnalu matrame kaligi untaayi. 0 lakshanam unte taruvaati sharatu ellappuduu santrupti chendutundi.
parimita galois podigimpu kosam, galois samuuham anedi kshetra automorphisms yokka samuuham, avi chinnavi (i.ii., adanamgaa mariyu gunakaaram samrakshinche mariyu tama muulakaalanu pampe bijections). eee samuuham yokka praamukhyata galois siddhaantam yokka praathamika siddhaantam nundi udbhavinchindi, idhi upa samuuhaala samiti mariyu podigimpu yokka madhyantara podigimpula samiti madhya spashtamaina okariki okaru gala paraspara sambandhaanni nirmistundi. eee correspondence dwara, group-theoretic lakshanaalu kshetraala gurinchi vaastavaalaloki anuvadistaayi. udaaharanaku, paina perkonna vidhamgaa galois podigimpu yokka galois samuuham parishkarinchalenidi ayithe (abelian samuuhaala nundi nirminchalemu), appudu "" yokka sunnalanu adanamgaa, gunakam mariyu radicals paramgaa vyakteekarinchalemu, i.ii., suutram _ 12 thoo kuudina vyakteekaranalu. udaaharanaku, samaroopa samuuhaalu parishkarinchalenivi. tatphalitamgaa, chuupinatlugaa, eee krindi bahupadaala sunnalu mottaalu, utpattulu mariyu radicals dwara vyakteekarinchabadavu. taruvaati bahupadi kosam, eee vaastavaanni abel-rufini siddhaantam antaru
kshetraala yokka tensor utpatti saadhaaranamgaa oka kshetram kaadu. udaaharanaku, degree yokka parimita podigimpu anedi-alzebras yokka isomorphism unte matrame galois podigimpu avutundi.
eee vaastavam grothendic yokka galois siddhaantaaniki prarambham, idhi alzebro-geometric vastuvulaku vartinche galois siddhaantam yokka suduura podigimpu.
kshetraala yokka asthiratalu.
oka kshetram yokka praathamika asthirathalo daani pradhana kshetram yokka lakshanam mariyu adhigaminche sthaayi untaayi. rendavadi pradhana kshetrampai beejaganitaparamgaa swatantramgaa unde muulakaala garishta sankhyagaa nirvachinchabadindi. eee rendu dayta angeekaristhe rendu beejaganitamgaa moosiveyabadina kshetraalu mariyu khachitamgaa isomorphicga untaayi. idhi oke cardinality mariyu oke lakshanam kaligina lekkinchaleni beejaganitamgaa moosiveyabadina edainaa rendu kshetraalu isomorphic anie suuchistumdi. udaaharanaku, mariyu isomorphic (conny topological feeldlugaa isomorphic kaadu).
kshetraala namuunaa siddhaantam.
namuunaa siddhaantamlo, ganita tarkam yokka oka saakha, rendu kshetraalu mariyu prati ganita prakatanaku kooda nijam mariyu dheeniki viruddhamgaa unte muulakamgaa samanam anie pilustaaru. prasnalooni ganita prakatanalu modati-krama vakyaluga undaali (0,1, adanamgaa mariyu gunanamtho kuudinavi). oka vilakshanhamaina udaaharana, oka puurnaamkam kosam,
anni vyakteekaranalaku atuvanti suutraala samiti beejaganitamgaa moosiveyabadindi.
lefshetz suutram lakshana sunna yokka edainaa beejaganitamgaa moosivesina kshetraaniki muulakamgaa samanamga untundani perkondi. antekakunda, edainaa sthiramaina prakatana taginanta adhika lakshanam kaligina edainaa beejaganitamgaa moosivesina kshetramlo unte mariyu unte matrame untundi.
oka sett meeda ultrafilter ayithe, mariyu pratidaanikii oka feeled ayithe, daaniki sambandhinchi ultraproduct anedi oka feeled. idhi suuchimchabadutumdi
idhi kshetraala parimitigaa aneka vidhaluga pravartistundi kabatti λος yokka siddhaantam, annintikee conny chaalaa varaku kaligi unna edainaa modati arder statement, ultraproduct kosam kooda kaligi untundani perkondi. paina perkonna vaakyaaniki vartimpajeste, idhi oka isomorphism undani choopistundi.
paina perkonna acks-kochen siddhaantam kooda dheeni nundi anusaristundi mariyu ultraproduct yokka isomorphism (rendu sandarbhaallo anni praimlapai)
adanamgaa, modal siddhaantam nijamaina closed fields ledhaa exponential fields (ivi exponential function kaligi untaayi) vanti vividha itara rakaala feeldla yokka taarkika lakshanaalanu kooda adhyayanam chestundi.
sampoorna galois samuuham.
beejaganitaparamgaa moosiveyabadani (ledhaa vaeru cheyaleni moosiveyabadani) kshetraalaku, sampoorna galois samuuham praathamikamgaa mukhyamainadih paina vivarinchina parimita galois podigimpula kesunu vistarinchadam, eee samuuham "anni" parimita vaeru cheyagala podigimpulanu niyantristundi. praathamika maargaala dwara, samuuhaanni proofer samuuhamgaa chuupavachchu, idhi samagramaina puurti. eee prakatana, beejaganita podigimpulu matrame eee parimita podigimpula yokka kshetraalu, mariyu eee parimita podigimpula yokka galois samuuhaalu eee vidhamgaa ivvabaddaayi anae vaastavaanni upasamharinchukuntundi.
generatorlu mariyu sambandhaala paramgaa vivarana anedi-adik sankhya kshetraala (parimita podigimpulu) yokka galois samuuhaalaku kooda prasiddhi chendindi.
langlands programme vanti ankaganitamloni aneka saakhalalo galois samuuhaala praatinidhyaalu mariyu wheel group vanti sambandhita samuuhaala praatinidhyaalu praadhamikamainavi. atuvanti praatinidhyaala yokka cohological adhyayanam galois cohologyni upayoginchi jarugutundi. udaaharanaku, saastriiyamgaa central simple-alzebras samuuhamgaa nirvachinchabadina bruyer samuuhaanni, galois cohomology samuuhamgaa tirigi ardham chesukovacchu, avi
keey-siddhaantam.
milnor keey-siddhaantam ilaa nirvachinchabadindi
2000 loe vladymyr vovodsky niroopinchina pramaanam avashesha isomorphism siddhaantam, dheenini isomorphism dwara galois cohomologyki sambandhinchinadi.
beejaganita k-siddhaantam icchina kshetramloni gunakaalato viloma maatrikala samuuhaaniki sambandhinchinadi. udaaharanaku, invertible matricks yokka determinantnu teesukune prakriya isomorphysaniki daariteestundi. matsumoto siddhaantam dheeniki angeekaristundani choopistundi. adhika sthaayilalo, k-siddhaantam milnar k-siddhaantam nundi vaerugaa untundi mariyu saadhaaranamgaa ganinchadam kashtam.
applications.
leanier beejaganitam mariyu commutative beejaganitam.
okavela, appudu sameekaranam
oka kshetramlo oka pratyaekamaina parishkaram undhi, anagaa suutram _ 16, oka kshetra nirvachanam yokka eee takshana parinaamam sarala beejaganitamlo praadhamikamainadi. udaaharanaku, idhi gassian tolagimpuloo oka mukhyamaina padaardham mariyu edainaa vector space oka aadharam kaligi undataaniki rujuvu.
modules siddhaantam (kshetraalaku badulugaa valayalapai vector spacela analog) chaalaa klishtamgaa untundi, endukante paina perkonna sameekaranaaniki aneka ledhaa parishkaralu undakapovachhu. nirdishta vyavasthalalo, ring meeda leanier sameekaranaalu fields vishayamlo kante parishkarinchadam chaalaa kashtam, mukhyamgaa integers ring vishayamlo kooda.
parimita kshetraaluh guudha lipi shaastram mariyu coding siddhaantam.
vistrutamgaa vartimpajeyabadina guudha lipi shaastra dinacharya vivikta ghaataankam anae vaastavaanni upayogistundi, i.ii., computing
oka (pedda) parimita kshetramlo vivikta lagarithm kante chaalaa samarthavantamgaa nirvahinchavacchu, idhi viloma operation, i.ii., oka sameekaranaaniki parishkaranni nirnayinchadam
deerghavruttaakaara vakra lipi saastramlo, parimita kshetramlo gunakaaram deerghavruttaakaara vakrarekhapai paayintlanu jodinche operation dwara bharti cheyabadutundi, i.ii., roopam yokka sameekaranam yokka parishkaralu
parimita kshetraalu coding siddhaantam mariyu combinatics loe kooda upayoginchabadataayi.
jyaamitih functionla feeled.
oka kshetramloki tagina sthalavignaana sthalamlo vidhulanu jodinchavachhu mariyu binduvulugaa guninchavacchu, udhaa.zii.rendu vidhula yokka utpatti domine lopala vaati viluvala utpatti dwara nirvachinchabadutundih
idhi eee vidhulanu-commutative beejaganitam chestundi.
functionla "feeled" kaligi undataaniki, integral domainlugaa unde functionla algibralanu pariganinchaali. eee sandarbhamlo rendu vidhula nishpattulu, i.ii., roopam yokka vyakteekaranalu
functionla feeled anie piluvabade feeldnu yerparustundi.
idhi rendu pradhana kesulalo jarugutundi. sanklishtamaina aneka retlu unnappudu. eee sandarbhamlo, okaru holomorphic functionla beejaganitam, i nii pariganistaaru.ii.sanklishtamaina differentiable functionlu. vaati nishpattulu meromorphic functionla feeldnu yerparustaayi.
beejaganita vaividhyam yokka function feeled (bahupadi sameekaranaala saadhaarana sunnaluga nirvachinchabadina rekhaganita vastuvu) saadhaarana functionla nishpattulanu kaligi untundi, i.ii., vaividhyampai bahupadi vidhula nishpattulu. oka kshetrampai-dimensional space yokka function feeled, i.ii., anischitulalo bahupadaala nishpattulatoe kuudina kshetram. function feeled edainaa open dense subwarityki samanamga untundi. maroe maatalo cheppalante, function feeled anedi (koddigaa) chinna upavargamtho bharti cheyadaaniki sunnitamgaa undadu.
function feeled isomorphism mariyu birational equivalence off virusies kinda asthiramgaa untundi. anduvalla idhi nairoopya beejaganita rakaala adhyayanam mariyu beejaganita rakaala vargeekaranaku oka mukhyamaina sadhanam. udaaharanaku, birational equivalence kinda, transpendence degreeki samaanamaina dimension anedi asthiramgaa untundi. vakrarekhalu kosam (i.ii.parimaanam okati), function feeled chaalaa daggaragaa untundih mruduvugaa mariyu sarainadi ayithe (compactga undatam yokka analog), daani functionla feeled nundi isomorphism varaku punarnirminchavachhu. adhika parimaanamlo function feeled takkuva gurthunchukuntundi, conny inca daani gurinchi nirnayaatmaka samacharam untundi. adhika kolatalalo function fields mariyu vaati rekhaganita ardhaanni adhyayanam cheyadaanni birational jyaamitigaa suuchistaaru. kaneesa namuunaa karyakramam suuchimchina function feeled thoo saralamaina (nirdishta khachitamaina ardhamlo) beejaganita rakalanu gurtinchadaaniki prayatnistundi.
sankhya siddhaantam prapancha kshetraalu.
beejaganita sankhya siddhaantam mariyu ankaganita jyaamitiloo prapancha rangaalu pramukhamgaa unnaayi.
avi, nirvachanam prakaram, sankhya kshetraalu (parimita podigimpulu) ledhaa function kshetraalu (parimita podigimpulu). sthaanika kshetraala vishayaanikosthe, eee rendu rakaala kshetraalu varusaga 0 mariyu saanukuula lakshanaalanu kaligi unnappatikee, aneka saaruupya lakshanaalanu panchukuntaayi. eee function feeled saaruupyata ganita anchanaalanu roopondinchadaaniki sahaayapadutundi, tarachugaa modata function feeldla gurinchi prasnalanu ardham chesukovadam dwara, taruvaata sankhya feeled kesunu pariganinchadam dwara. rendodi tarachugaa marinta kashtam. udaaharanaku, reeman gta function yokka sunnalaku sambandhinchina reeman parikalpana (2017 natiki terichi undhi) wheel oohalaku samaantaramgaa pariganinchabadutundi (1974loe pier dilligne dwara niroopinchabadindi).
cyclotomic kshetraalu atyanta teevramgaa adhyayanam cheyabadina sankhya kshetraalalo okati. avi roopaaniki chendinavi, ikkada aikyata yokka aadima-va muulam, i.ii.sanklishtamaina sankhya santruptikaramgaa mariyu andarikee. regular praimga undataaniki, kumar fermat yokka chivari siddhaantaanni niroopinchadaaniki cyclotomic feeldlanu upayoginchaadu, idhi sameekaranaaniki hetubaddhamaina nanziro parishkarala unikini nokki chebutundi.
sthaanika kshetraalu prapancha kshetraala puurti. ostrovsky siddhaantam, oka prapancha kshetram yokka puurti matrame sthaanika kshetraalu mariyu kshetraalu anie nokki chebutundi. prapancha rangaalalo ankaganita prasnalanu adhyayanam cheyadam konnisaarlu sthaanikamgaa sambandhita prasnalanu chudatam dwara cheyavachu. eee saanketikatanu sthaanika-prapancha suutram antaaru. udaaharanaku, hassey-minkovsky siddhaantam eee sameekaranaalanu parishkarinchadaaniki chaturasra sameekaranaala hetubaddhamaina parishkaralanu kanugone samasyanu taggistundi, dheeni parishkaralanu sulabhamgaa vivarinchavacchu.
sthaanika kshetraala maadirigaa kakunda, prapancha kshetraala galay samuuhaalu teliyavu. viloma galois siddhaantam edainaa parimita samuuham konni sankhyala kshetraaniki galois samuuham kaadaa anae (parishkarinchani) samasyanu adhyayanam chestundi. class feeled siddhaantam abelian podigimpulanu vivaristundi, i.ii.abelian galois samuuhamtoe, ledhaa samanamga global fields yokka abelionized galois samuuhaalatoe. kronekar-webber siddhaantam anae shaastreeya prakatana, garishta abelian podigimpunu vivaristundihi idhi kshetram
aikyata yokka anni aadima muulaala dwara pondabadindi. kronekar yokka jugendtram saadhaarana sankhyala kshetraala yokka adhe vidhamaina spashtamaina vivarananu adugutundi. oohaatmaka chaturasra kshetraala kosam, suutram _ 18, sanklishta gunakaaram siddhaantam deerghavruttaakaara vakrarekhalanu upayoginchi vivaristundi. saadhaarana sankhyala kosam, atuvanti spashtamaina vivarana teliyadu.
sambandhita bhavalu.
kshetraalu aasvaadinchagala adanapu nirmaanaaniki adanamgaa, kshetraalu vividha itara sambandhita bhaavanalanu angeekaristaayi. edainaa feeled loe 0 ≤1 unnanduna, edainaa feeled loe kaneesam rendu elements untaayi. edemaina, oka muulakamtoe kuudina kshetram yokka bhavana undhi, idhi parimita kshetraala parimitigaa soochinchabadindi, 1kii untundi. vibhajana valayaalatoe paatu, quasifields, sameepa-kshetraalu mariyu semifields vanti kshetraalaku sambandhinchina aneka itara balaheena beejaganita nirmaanaalu unnaayi.
kshetra nirmaanamtho saraina taragatulu kooda unnaayi, veetini konnisaarlu kshetraalu anie pilustaaru, veetilo f pedda aksharam untundi. adhivaastavika sankhyalu vaastavikata kaligina kshetraanni yerparustaayi, avi oka samiti kakunda saraina taragati anae vaastavam tappa oka kshetram avutundi. game siddhaantam nundi vacchina nimbers anae bhavana kooda atuvanti kshetraanni yerparustundi.
division ringulu.
oka kshetram yokka nirvachanamlo okati ledhaa aneka suutraalanu vadulukovadam itara beejaganita nirmaanaalaku daariteestundi. paina paerkonnatlugaa, commutative ringulu multiplicative inverse uniki minaha anni feeled suutraalanu santrupti parustaayi. badulugaa gunakaaram yokka commutativityni tagginchadam anedi "division ring" ledhaa "sque feeled" anae bhavanaku daariteestundi; konnisaarlu associativity kooda balaheenapadutundi. parimita-dimensional-vector khaaliilu ayina ekaika vibhajana valayalu, (idhi oka kshetram), mariyu quaternians (dheenilo gunakaaram naan-commutative). eee phalitaanni frobenius siddhaantam antaaru. octonianlu, veeti kosam gunakaaram anedi commutative ledhaa associative kaadu, idhi oka praamaanika pratyaamnaaya vibhajana beejaganitam, conny vibhajana ring kaadu. eee vaastavam 1958loe maikhel kerwire, raul bottle mariyu zaan milner dwara beejaganita topology paddhatulanu upayoginchi niroopinchabadindi. besi-dimensional division beejaganitam yokka uniki lekapovadam marinta saastreeyamainadi. kudi vaipuna unna ventrukala banthi siddhaantam nundi dheenini oohinchavachhu.

