in Galois Theory What's ? Next
finite extension well understand super We infinite degree extensions ? what about no algebra " hare " extensions " purely transcendental " → " extensions LOTS at algebra have algebraic purely " → " Galois Theory " for is There Good news : extensions algebraic infinite complicated it's Bad more news :
degree over Q ( lip :p is prime ) ) EIK is intuit Gall Kla ) so big ! is correspond to degree of of 2 extensions Do Q index that 2 Gal ( Kla ) of are subgroups . 2 in Gall Kla ) ⇒ NO Gall HQ ) and Nis index Gul ( 14¥ ' Iz . counted N such are But by surjeotieexGnlll4@7-n7Lz.B - t
surjection vncountubly such many There . are 2 extensions of Q ? uncountable degree Are the many : only ! countably may No and index 2 subgroups So : too may Q of degree extensions 2 enough not . then focus and Gal ( Kia ) topolegize Resolution : " topologically subgroups that " nice are on
, then Gall HF ) infinite degree If KIF is also " pntinik it's " " topological " , but not a is on of finite groups " inverse limit " group , . ie , an - f - a- finite EI Kp # Kp . ← Kpk - - - " " " . . . . RHYME T oi , " . Fortuny { ioaipi ai Ekp ) : groups of Galois of the limit inverse finite extensions of F .
questions two big There are - " separable the F' set Galois groups ) let be ① ( Absolute extension et F that's separable of ( smallest " closure F closed ) - Gal ( FTEPIF ) algebraically . Define Gp . Galois They understand : if Infinite GF says you completely understand I completely , then you . is really really scary GF Sad news : . well . Ga we don't understand very example For :
questions related Some : " ' ' special groups of form GE away how ⑨ are pnetrnik groups ? class of the larger Grothendieck ) Gia ? ( see ⑥ what is Q does for a given group ④ , ( appropriately topologically ) GE → Q have : equivalent Galois Thang F : does to By Gal ( KIFKQ , extension w/ KIF an
② gives Problem Galois " Inurn " to rise a given field and BigQu#2 If F is find KIF we a given can group G is , en ? Gall KIF ) t so Then with IFI - o a field be I let F . for Ge In iff " " G realizable F is over some n .
about it Q ? F What - - , but completely understood not Q realizable → any over Sn is realizable over is solvable → group . every - - how G , can and F group given For with a k extension for searching an go we Gal ( KIF ) ' G ? induction idea :
with % a Q HOG has G some Suppose . with Galois Thy , if had KIF some we By we'd have then FELEK Gal ( KH a G , Gal ( TF ) a- Od with Gul ( Kk ) IN and . ± :i÷:÷÷ :* ::÷:÷¥ : found - extension we Q " is Galois " big extension ^ ③ make sure this " glue correctly " to G and Q N and
, if with Gallen )=K instance FELEK For Gal ( YF ) ' Ezio Ka , then and K l Ka of God ( " le ) be will one L Re Zz ④ Be Zettel Ky @ Kz or F 174 Qg or or " Galois This - related to strategy is " problems embedding
of my research connected to Then some . are Gul LKIFKQ up Ek item : suppose and Basic . Kummer Theory I :÷÷ : :¥i :* ! ) : " :i÷ . . it - Ciri . if ( Fyi -
vision of research , I study relative In a my Theory Kummer : I :¥÷÷÷÷:÷ ⇒ 3 viii. iii. I - I a' era ) { IE ri : .
⇒ " Galois modules " Resounding Theme These are : " stratified random " Than fur a more would module be . differ from random much Gf is prelim 't group .
to able G- and I Heller Lauren were when churl F) =p structure compute Kp ④ Kp Gul ( KIF ) ' and .
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