II Relative : tasting Galois computation of An a explicit - PowerPoint PPT Presentation

Galois Group The Galois theory II Relative :

tasting Galois computation of An a explicit group

Newstead E ' FEKEE , how If Gul 'Elk ) ÷ :÷÷¥÷ : " " " . . . . other ? each te

subgroups ) Lemma ( Tep sub extensions are IF FE KEE , Gal ( Elk ) EGAICEIF ) then . Gall Elk ) C- Gull Elf ) check PI to need we . with re Aut CE ) regal ( Elk ) implies Now = ideate = folk ) If - idf Hence off - idk Mk . ' . it Gal ( Elf ) Therefore . IDK

" also " bottom sub extensions Nhlvral question : are subgroups ? No . = TE Gall KIF ) , do we E If TE Gal ( EIF ) ? I have k - K -5k , and Recall T Ip } ' TEGAKKIF ) defined isn't it on hence of E . all

element of rt Gul ( EIF ) Can make an we Gal ( KIF ) . en E , not k . a fucken Notimnediatdy is : r : K → E 4k know do However , we . element of be for rlk to an order In ion ( rha ) - K need Gall KIF ) - , we . happen ? this When does

field splitting Neaexampte the het be E ( Vz ) x ' -2 , for and let K - - . Q1 ) ( Here F - - : . with TE Gall Ela ) take Suppose we = as rlxz ) and old , ) =L , . homework k ? imlrlk ) we from = Is - * * ④ (a) - K know da ) - - .

" ) " nice Thy ( when restrictions are the splitting field of where k is FE KEE , If the splitting field of gcx ) c- FIX ) is and E re Gal ( EIF ) all for e F EXT , then f- Ix ) im ( Nk ) K have - - . we of glx ) roots the PI let . . , pmek be . pi , ' - - Palm seen then that { 13 , : a- eiadlirfep . " we've F - basis for k . is an

. pm } { is . . permutes key fact : r . . . . im ( Hk ) E K First let's show . - - ' Prem . . - em Pie - Z te ' be k given KEK so Let - . , , ' - - Bim ) = r ( Efe . Observe . . - em Pie ruck ) = Efe . - - r ( Pm ) " E K . . - em r ( pile ' If I . " 2 " resolves argument Hence in ( Hk ) Ek . Similar . ④

( Restrictions to splitting " nice " ) field Core are field The splitting k where is If FE KEE is th sphlty field and glx ) EF Ix ) E for : Gallet F) → Gal ( KIF ) then f ffx ) EF Ix ) , far homomorphism = rly YG ) is a by given = Gal ( Elk ) Kerl 4) with .

. Operation - defined PI well 4 know is we , Ik k Ik , re Ik o preserving is r = . - idk } = { TE Gall EIF ) : Hk Ker (4) - Now = { re Aut ( E ) = idk } : elk Gal ( Elk ) = . FE

Quotients ) Core ( Galois is th splitting field for k where FEKEE If is the splitting field for E c- FIX ) and separable gcx ) Gul ( Elk ) a Gullet ) f- Ix ) c- FIX , then separable Gal ( Elf ) Kal CE Ik ) Gal ( KIF ) . = and that y from the check we only need to PI surjective result is last .

- IE : F ) I Gul ( EIF ) I and know We - - FT = [ K ( Gul ( KIF ) I separable ' . splitting field fernflxleklx ) the E is Observe that , we get = [ E : KJ . So I Gul ( Elk ) I and - Y;:Y¥i - I' cent " " " ' limits - - - - = [ E : KIK : F ) = ( Gal ( HEH I - I codomain Ik : F) - = T enay