abstract algebra

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1、CHAPTER1NumberFields1.Example:QuadraticnumberfieldsBeforeweconsidernumberfieldsingeneral,letusbeginwiththefairlyconcretecaseofquadraticnumberfields.AquadraticnumberfieldisanextensionKofQofdegree2.Thefundamentalexamples(infact,asweshallseeinamomenttheonlyexample)arefieldsoftheform√√Q(d)={a+bd

2、a,b∈

3、Q}whered∈Qisnotthesquareofanotherrationalnumber.Thereisanissuethatarisesassoonaswewritedownthesefields,anditis√importantthatwedealwithitimmediately:whatexactlydowemeanbyd?Thereareseveralpossibleanswerstothisquestion.Themostobviousisthatby√√dwemeanaspecificchoiceofacomplexsquarerootofd.Q(d)isth

4、endefinedasasubfieldofthecomplexnumbers.Thedifficultywiththisisthatthenotation√“d”isambiguous;dhastwocomplexsquareroots,andthereisnoalgebraicwaytotellthemapart.Algebraistshaveastandardwaytoavoidthissortofambiguity;wecansimplydefine√Q(d)=Q[x]/(x2−d).√Thereisnoambiguitywiththisnotation;dreallymeans

5、x,andxbehavesasaformalalgebraicobjectwiththepropertythatx2=d.Thisseconddefinitionissomehowthealgebraicallycorrectone,asthereisno√ambiguityanditallowsQ(d)toexistcompletelyindependentlyofthecomplex√numbers.However,itisfareasiertothinkaboutQ(d)asasubfieldofthecomplex√numbers.TheabilitytothinkofQ(

6、d)asasubfieldofthecomplexnumbersalso√√becomesimportantwhenonewishestocomparefieldsQ(d1)andQ(d2)fortwodifferentnumbersdandd;theabstractalgebraicfieldsQ[x]/(x2−d)and121Q[y]/(y2−d)havenonaturalrelationtoeachother,whilethesesamefieldsviewed2assubfieldsofCcanbecomparedmoreeasily.Thebestapproach,then,se

7、emstobetopretendtofollowtheformalalgebraicoption,buttoactuallyvieweverythingassubfieldsofthecomplexnumbers.Wecandothisthroughthenotionofacomplexembedding;thisissimplyaninjectionσ:Q[x]/(x2−d)/→C.Aswehavealreadyobserved,thereareexactlytwosuchmaps,oneforeachcomplexsquarerootofd.Beforewecontinuew

8、ereallyoughttodecidewhichcomplexnumberwemean√byd.Thereisunfortunatelynoconsistentwaytodothis,inthesensethatwe561.NUMBERFIELDScannotarrangetohavepppd1d2=d1d2√foralld1,d2∈Q.Inordertobeconcrete,letuschoose√dtobethepositivesquarerootofdforalld>0anddtob