Lattimer subsets of Art IE ) field function for Fixed & sub - PowerPoint PPT Presentation

of Characters Independence fields Fixed of I : subgroups

Lattimer subsets of Art IE ) field function for Fixed & sub extensions of E subsets et Aut ( E ) lie , inclusion reversing ) " contravariant " → not too small ) ( fixed feild , : EG ) 3141 → [ E . - son ) 99 . . . , rn ) EAT LE ) , then tool : if Er . . necessary - independent " E " are

NYn we stty Wut happens when of fields subgroups fixed : EG ) and about IE more say → we can IGI ? function field fixed " ? " injective is → subgraph on

her subgroups ) matches ( Bigness theorem = IGI IE : EG ) GE Atle ) . Then . Suppose > IGI : EG ) CE Wc already PI know . CE : EG ) > 1Gt contradiction . assume For , 1Gt - n . . , rn ) , so - Er , . G . write lets - . . , Ent . ) E E have { e , . assumption we By - independent EG . that are

linear system of equations : we'll create a = O , )xnt , legit Tile ) x , t - to { - - ; L = = 0 lentil xnt , race , )x , t - t - . than equations . £ in variables , there more Since are solution - trivial he . has non know a we number of elements nonzero minimal The be Let r ye solution to nontrvinl in . a

r may " nice solution " with to L create let's a components . nonzero we have columns of Lo assume can , we rearranging By with all xito . - - yo ) ( x , , of term . . ,Xr , O , solution . . a solution still produces Scaling by ' a Xi , o ) gi to with all ( l , ya , . , yr , q . . - - . . w ' Xzxi idq to see corresponding we on the Focus , row

id Cena ) yay . - 0 idle , ) Lt idler ) yzt - t - - - indepef leg . > em ! EG git EG violates , This all If Ef EG has yi some yi So : yr # EG assume columns can . After carrying , with ri EG exists some Hence , there rilyr ) t yr .

of each create L act row a let to on ri oil system new : { to , lentil xnti ) =D tri ( T , (4) Xi t - - i. grid - t Talent , ) Xna ) =D Oi ( Tale , )x , t - = { - - trio , lent . ) rilxnti ) Orion circle . )oi( x , ) t ; = - - t rirnlenti ) rilxnti ) =D ( e.) ri (a) t . . .ir/xutiDsohesriL iff ( rlx . ) . solves L . - . xut . ) ① ( x . , . . . , on ) . - i , rion ) :{ r , , GEAVHE ) Erin , ② since ,

iff solves L . , Xu , ) ( x . . have Hence we . L solves . . . , rilxnti ) ) ( rilx . ) , particular In rilyr ) , rill , rill ) . - ie ) - ( rill ) , oily . ) , ( l , ya , - - syr ,0 . - yo ) - oily ) , - oilyr ) , o , ⇐ ( O , ya . . , gr . . - to since L solves oil yr ) # yr . too few has terms - zero TIX It non . → ←

Ouranginalypictre G sub extensions of E HH et Aut ( E ) subgroup ' . contravariant properties " bigness " • preserves

Cos ( Fixed field function is injection on subgroups ) 't = EG , then H=G . have , Ge Aut CE ) E If it E " " = E " " PI - EG " - E gives E - claim - " ) " fun exercise prove This ( you as can " " ) = ( E > I Hv G) 't ) htt = IE : E : E So : but reverse same argument Run the GE H So . we get HEG H & G , roles of the µ .

Ouronginalypictre G sub extensions of E HH et Aut ( E ) subgroup ' . contravariant properties " bigness " • preserves • injective