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cmcsc10Abstract.\`W*eDproveEcertainweakEversionsofEsomecelebratedresultsduetoAlexander $ Vishikdcomparingdrationalityofalgebraiccyclesoverdthefunctioneldofaquadricand$ overthebaseeld.
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cmmi10u-invqariantm*2^ 0er cmmi7ra+m1(forry3)isextendedtoarbitrarycharacteristic$ 6=2. XQ cmr12ThemainresultsofthisnoteareTheoremhtml:1.1 html:(thebasicresult)withitsenhancemenrthtml:2.1 html:,PropSositionhtml:3.1 html: withitsenhancemenrthtml:4.1 html:implyingTheoremshtml:3.2 html:andhtml:3.3 html:(whicrhgoalittledbitdbSeyrondthebasicresult),and(aquitespSecial)Propositiondhtml:5.3 html:(goinginaspSecialsituationevrenmorebSeyondthebasicresult).8ThemainapplicationisTheoremhtml:5.1 html:. Inbcrharacteristicc0,QallofthishasbSeenprorvedseveralbyearsagobbyAlexanderbVishikine[html:3 html:]fand[(html:5 html:](exactfreferencesaregivrenrightebSeforeeacrhstatement)ewithahelpofthealgebraiccobSordismtheoryandespecially2@ cmti12symmetric
opfferationsof[html:4 html:].=Infact,ytheoriginalʜvrersionsoftheʝmostresultsarestronger.ؼTheydonotinrvolveʜtheassumptionthat&thegroup&CHr&(x#g cmmi12Y m)(notationinrtroSducedin&thebeginningofSection&html:1 html:)is2-torsion-free(andthereforehasno2-primarytorsion),madehereinTheoremhtml:1.1 html:andPropSositionhtml:3.1 html:.Ourvrersionswiththe2-torsion-freeCHB(xY m)(letuscallthemvery>Jweffak)areevenweakerthan1`thewreakversions1_of[html:3 html:,CRemarkonPage370]where,Croughly1_spSeaking,for1`YйwitharbitraryCHf4(xY m)theresultsareobtaineduptoanelemenrtofexpSonenrt2inCHf5(xY m)andwhicrhj(theweakversions)canbSeobtained(stilljincharacteristic0only)withahelpoftheLandwrebSer-Novikovoperations(stillinthealgebraiccobSordismtheory)replacingthesymmetricopSerations. AlthoughJthevreryweakversionsarealreadysucientforallexistingapplications,
3weprorve)thewreak)versions)aswrell(seeTheoremhtml:2.1 html:andPropSositionhtml:4.1 html:).Theproofshereare!only!abitmorecomplicated(thaninthevreryweakcase)and!haveanadvXantage:theyarvoidv6inductionv7bydimensionofthev7quadrichtml:Y!|{Y cmr81@ html:andthereforecanbSeadoptedtoservreforthetproSofofPropositionhtml:5.3 html:ofthelasttsection,Pwheredimensionofthequadricisspecic.PropSositions:html:5.3 html:isthenals9stepinextendingconstructionofeldsofu-inrvXariants:22$2 cmmi8r+1foranryr&!",
cmsy10UR3givenin[(html:5 html:]forcharacteristic0(thecaseofr=UR3hasbSeendoneearlierand
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