
saral vistaar

kshetra siddhant mein, ack saral vistaar ack kshetra vistaar haye joe ack ekal tatva key sanyojan dwara utpann hotaa haye, jise "adim tatva" kahaa jaataa haye. saral vistaaron koo achchi tarah seey samajha jaataa haye aur inhen poori tarah seey vargeekrit kiya jaa sakata haye.
adim tatva pramey seemit saral vistaaron kaa ack charitra varnan pradaan kartaa haye.
paribhasha.
ack kshetra vistaar koo ack saral vistaar kahaa jaataa haye yadi "l" mein ack tatva maujood haye
iska matlab haye kii pratyek tatva koo ack tarksangat ansh key roop mein vyakt kiya jaa sakata haye, jisamen gunank shamil hain.
dou alag-alag prakaar key saral vistaar hain.
tatva pardarshi hoe sakata haye, jiska arth haye kii yeh kissy bhee bahupad kaa mool naheen haye jisamen gunank hain. iss mamle mein sootra _ 2 tarksangat karyon key kshetra key liye samroop haye sootra _ 3
anyatha, beejganitiya haye; arthaat, ack bahupad kaa mool haye. nyuntam degree key monic bahupad sootra _ 4 koo, mool key roop mein, nyuntam bahupad kahaa jaataa haye. isaki degree kshetra vistaar kii degree key barabar haye, yaani-vector sthaan key roop mein dekhaa jaane vaalaa ayam. iss mamle mein, sootra _ 2 key pratyek tatva koo vishisht roop seey ack bahupad key roop mein vyakt kiya jaa sakata haye joe degree seey cumm haye, aur sootra _ 2 bhagfal ring sootra _ 7 key liye samroop haye.
donon hii mamalon mein, tatva koo vistaar key liye ack utpann karane vaalaa tatva yaa adim tatva kahaa jaataa haye; ack kaa kahana haye kii yeh bhee utpann hotaa haye.
udaaharan key liye, pratyek parimit kshetra usee visheshta key pradhaan kshetra kaa ack saral vistaar haye. adhik sateek roop seey, yadi ack abhaajya sankhya aur sootra haye tou tatvon kaa kshetra sootra _ 9 sootra _ 10 kii degree kaa ack saral vistaar haye, iska matlab haye kii yeh ack tatva dwara utpann hotaa haye joe degree key ack aparivartaniya bahupad kaa mool haye. 
haalaanki, seemit kshetron key mamle mein, aamtaur parr "adim tatva" key roop mein sandarbhit naheen kiya jaataa haye, bhale hii yeh oopar dee gaee paribhasha key anuroop hoe.
iska kaaran yeh haye kii seemit kshetron key mamle mein, ack adim tatva kii ack pratispardhi paribhasha haye. vaastav mein, ack parimit kshetra kaa ack adim tatva ack aisaa tatva haye kii kshetra kaa pratyek gair-shoonya tatva ack shakti haye; yaani, kshetra key gunak samooh kaa ack generator (vivaran key liye dekhein aur adim tatva (parimit kshetra)). ack adim tatva (iss arth mein) ack utpann karane vaalaa tatva haye (kyonki ack purnank shakti ack bahupad kaa ack vishesh mamla haye), lekin vyutkram galat haye.
sankshep mein, ack saral vistaar kii samanya paribhasha key liye aavashyak haye kii kshetra key sabhi tatvon koo ack ekal "generator" mein bahupadon key roop mein vyakt kiya jaa sakata haye, jise "adim tatva" bhee kahaa jaataa haye. parimit kshetron key mamle mein, pratyek vistaar saral haye, aur sabhi gair-shoonya tatva ekal tatva kii shuddh shaktiyaan hain jinhen "adim tatva" bhee kahaa jaataa haye. inn arthon koo seemit kshetron key kshetra mein alag karane key liye, aam taur parr pahale arth key liye "generator" shabd kaa upayog kiya jaataa haye, aur ack dusare arth key liye "adim tatva" koo aarakshit kartaa haye. pahali dhaarana key liye kshetra adim tatva key shabdon aur dusare key liye samooh adim tatva kaa kabhi-kabhi upayog kiya jaataa haye.
saral vistaaron kii sanrachana.
yadi "l" "θ" dwara utpann "k" kaa ack saral vistaar haye tou yeh sabse chhota kshetra haye jisamen "k" aur "θ" donon hotey hain. iska matlab haye kii "l" key pratyek tatva koo "k" aur "θ" key tatvon seey parimit roop seey kaee kshetra sanchaalan (jodd, ghataav, gunan aur vibhajan) dwara praapt kiya jaa sakata haye.
bahupad valay "k" ["x"] parr vichaar karein. isake mukhya gunon mein seey ack yeh haye kii ack advitiya ring homomorphism maujood haye. 
dou mamle hoe sakate hain.
yadi sootra _ 12 injection haye, tou ise "k" ["x"] key ansh "k" ("x") key kshetra taka badhaaya jaa sakata haye. jaisa kii hamane maanaa haye kii "l" "θ" dwara utpann hotaa haye, iska tatparya haye kii sootra _ 12 "k" ("x") seey "l" parr ack samroopata haye. iska tatparya yeh haye kii "l" kaa pratyek tatva "θ" mein bahupadon key ack aparivartaniya ansh key barabar haye, aur aise dou aparivartaniya ansh samaan hain yadi aur kewal tabhi jaba ack ansh aur bhaajak koo "k" key samaan gair-shoonya tatva seey gunaa karke ack seey dusare mein paarit kiya jaa sakata haye.
yadi sootra _ 12 injection naheen haye, tou "p" (x) koo isake kernell kaa generator honey dein, joe iss prakaar "θ" kaa nyuntam bahupad haye. sootra _ 12 kii chhavi "l" kii ack upa-valay haye, aur iss prakaar ack abhinn kshetra haye. iska tatparya haye kii "p" ack aparivartaniya bahupad haye, aur iss prakaar bhagfal valay sootra _ 16 ack kshetra haye. jaisa kii "l" "θ" dwara utpann hotaa haye, sootra _ 12 adhikshep haye, aur sootra _ 12 sootra _ 16 seey "l" parr ack samroopata koo prerit kartaa haye. iska tatparya haye kii "l" kaa pratyek tatva "θ" mein ack advitiya bahupad key barabar haye, joe vistaar kii degree seey cumm degree kaa haye.

