Extension , Galois Galois Intermediate I : Extensions
lame l Gal LEIF ) t.EE if ] ' tf Galois Elf is :÷÷i÷÷ extensions ) characterizations for Galois Thin ( Equivalent - Gal ( EIF ) . Then and [ E : F) so let G - suppose equivalent following : are the :÷÷÷÷÷÷÷÷÷÷÷÷÷¥÷÷÷÷ Galois (4) Elf is
long time ago : .it#i::iEii:::D Quotients ) me and Elf are Galois Cory ( Galois ÷÷÷÷÷:¥÷÷÷ " Gal ( E # Kaye , k , Gall KIF ) . and = WTNwHti , ' ' iff " ? this Galois ? intermediate extensions when are
intermediate fields ) ( Conjugate ' Def n and and Tet FEKEE Galois be supper Elf , isomorphism exists there some . If FE KE E we say , Then Tle - side That T : k → I so conjugate to k . I is an equivalence relation ) " is " conjugate to Thin ( " I is conjugate to Kai by The given relation relation quintana . " is K an Through The motions . PI Go
for conjugate fields ) Thin ( Alternate description - { I : I conj ( K ) and let Galois be - let EIF , K ) Then conjugate to is . retral ( EIF ) } = { t ( K ) conj ( K ) : - . we hat regal LEIF ) , " Z a. k fr PI For " ay , - idp . with ( Mk ) Ip an isomorphisms - : K -7 idk ) 4k is : ktk with to k conjugate T II be ' ' e " . , let For Galois Elli we and are Elk By Thur 51 , since . So rlk ) - TWI re Gal LEIA extends to some know . t . 1B
- { K ) , Then - k HK ) have So : If conj ( K ) - we - re Gal ( EIF ) fr all . and of XT splitting field let x' - Le be Ex let E , - ⑥ ( WEE ) - ④ ( wt ) Ks and - k , - ④ ( Vz ) and ka - spaces They are as vector degree 3 so Each our is fields ? yes , Are Thy isomorphic as isomorphic ? . : arr ) → ④ ( wer ) build f " ' f can that we know we f root sending Vc to any by - 2) ' x' -2 Ff Q Q of idfirra 1352 ) ) . idk
⑥ ( w 't ) build 4,3 ( Va ) → Likewise can we : ④ → Q extends idea : that . , Ks } = { K Ceuj ( ki ) , .kz have Hence we D ? " " related to Galois ness Conjlk ) Hew is says if " Galois quotients " E. g. our , Gall Elk ) # Gal LEIF ) Galois , Then KIF is
⇒ intermediate Galois ness ) Chandu Eakins of Thy ( Equivalent and . Then let FE KEE Galois , be EIF let are egvinlent following . the H conj ( K ) = { K ) rly C- Gull Klf ) have ( " for of Gal ( EIF ) , all we ( iii ) Gal ( Elk ) A Gal ( EIF ) ( iv ) Khs Galois . is ' " happiness square of tf " use (a) ⇐ Ciii )
4) ⇒ Cii ) - ( K } ( oujlk ) Given : c- Gal ( KIF ) . 4k have foetal LEIF ) want : , we k to Hk ) them isomorphism always tha Note : is an show to We just need pointwise F fixes that . rt Gal LEIF ) . But all for HK ) - K : rt Gal LEIF ) ) = lock ) - Coujlk ) 1¥ { K ) . -
Ci ) ⇒ Ciii ) the C- Gall KIF ) have firebrat LEIF ) we Given : : Gull Elk ) s Gal ( Elf ) want have f : Gul LEIF ) - ' GUICHE ) hypothesis . we By our homomorphism Xlr )=Hk by . is a given : olk ' idk } Kerl 4) = Lothal LEIF ) Further = Gaulle Ik ) . Gal LEIF ) . Gal ( Elk ) o 305 , By Dad
Liu ) Ciii ) ⇒ Gal ( Elk ) s Gal LEIF ) Given : Galois . want KIF is : irreducible pile Fck ) i for all prove this by showing we'll and splits separable k , then plx ) with is root one m we already know Galois , in KUD . is Since Elf plus ) is , then root E has in ) plx Simon a roots E all in Leprnble and are . irreducibly Assume t th carthy that plxl has a least 2 at degree KID of fucker . in KCH , about in knew the root be we let a
at the - linear fuhr be mob let 13,8 and non By separability a. 13,8 KCXT . are of plx ) in , know Galois , we distinct . Elf Since all is - p that ok ) TE Gul LEIF ) - so there is Sean . there is and so Galois Elk is 9 , hwk By , with - 8 Tlp ) TE Gall Elk ) - . some know Gul ( Elk ) o Gal ( EIF ) , we assume Since we have pe Gal CEIK ) , we lothal LEIF ) and for ay fixes K . ' - ' e Gal ( Elk ) particular , 9pct . In 4pct
" ) should fix all - ' KEK Consider - ' Te ( o It r . . " 5) Ca ) - ' T @ we should ( r have AEK , Since - a . - - 1) (d) - ' ( t ( ok ) ) ) - ' t ( r " ) ( r Nate r = - ' ( t LPD = o - ' (8) - r - * a r (a) =p , last inequality fellows swim the where - ' ( p ) - d → a- so - r .
Civ ) ⇒ Ci ) Galois is KIF Given : = { K ) . conj ( K ) want : - phis 'm TE Gul ( KIF ) isom all is fer an that Note SF . extends idf : F - that from k to k TEGAILKIF ) each know for 51 Theorem we By of , E -7 E extensions r : [ Eik ] may have we an element of Gal LEIF ? extension such is Evey T take this form ? . Gal LEIF ) demark at How my - LE : F ) - IGNKEIFH = ( K : FILE : K ) I Gul ( Klett CE :k ) - # of extras choices for T
extension at retral ( Elf ) Since eade is an - K . HK ) have TE Gall KIF ) - we some , - { K ) Coujlk ) So - . 1¥
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