Applications . of Galois Proof ' TI : . Great Theorem
÷÷:÷:÷÷÷÷÷:÷÷÷ ' :÷::i::÷i÷÷::÷:::::i ( solvable group ) Deth and Hit YA ; cyclic is His Hit , fer all . so that Osian
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need we'll Thu ( Kummer Theory ) root of It nth primitive contains a and F NHN suppose hat EIF extension for unity Then an Wp . The splitting field for iff Gall EIF ) ↳ In E is E F EXT " some - c x . " next time " A we already . We 'll do know " E- " Nate . :
Great theorem ) Pf f Galois ' is solvable by . we'll Tet radicals f- KIEF fx ) 's first assume a solvable Gal ( Kf / F ) group prove That is . try to containing Kf radical tower : By assumption have a we Kf - . ← E - - → Es . , et Es F -7 E , → Ez ↳ - H 've ' Eire Est Ves ) H . ) . ) a @ ' II ni ) " aotunftg we'll extend picture by adjoining this .
T 'T ' " " " i seven . - - → Es . , et Es ← E > E , → Ez - F - - H 've Eire ) ' Est Ves ) H . ) nith field primitive F has Nate : fr all a kiss , the Hence fr all i - - ns ) - - ni - Min unity - ( ni of root w . subgroup of Rni Chinle Gull Ei/Ei . . ) get is a we Kummer theory cyclic ) by .
solvable Gal ( EIF ) strategy try to is prove Our : Gal ( Ket ) solvable order to is in ague . That Galois get ( Recall kffp we since is : , Gull ETF )/ga , ( E , µ , Ga ' ( KHF ) I . said last time Theorem from deep ish Our . ) solvable at solvable quotients groups are
So : we'll try Gal ( EIF ) solvable show to is . = Gall ETE ) E We'll fist focus on - . exercise : Eli . ) Galois both EIF ( Fun and are : Es Gall EIF ) Note . We'll show I solvable is . - Gal ( EYES ;) example So , fer Hi let . Gul ( El Es ) . Gal LEE ) :{ e ) Ho - . - .
get then we - EG.IE/E.)eGalIE/E . ) Ese )E Gal ( EYES . . ) E Gall { e ) e . . " t ' " " y Ho H , - I E E Hs Hz E Hs . . s - E - - - iff fundamental Theorem Hi o Hit , The says Es - YES , . , Kummer But extension Galois is a . Gul ( Es - YES , . . ) , and Galois it tells is even thy us cyclic ) subgroups of hence ( and Eni . But not are
Gull Es - i Hint - Gul sites . . . ) ' = yE ⇐ . . . know cynic which is we . solvable CT So : is . solvable , note Gal ( Eff ) is show Now to correspondence Galois , and by Galois F- ' Ftw ) IF so is Gall ETE ) o Gal ( Elf ) get we . 11 I
normal subgroup with we set : from work so previous ent¥¥, . ) c. Gal # E.) EGAHEIF ) Ese )e - e Gal ( EYE Gal LELE , ) e Gall le ) e . - " l ' " " y Ho H E - I E Hs Hz E Hs . . , E - - E - - " of this " lower layers satisfy The chain all Siwa layer top " property , and The " solubility since new Gal LEIF ) condition , have satisfies the also we solvable is .
half : second the For , and solvable Gal ( Kf IF ) we is assume medicals . solvable f by want is N ! th of unity . be - [ Kf : F ) and let a let w N - , If = Kf ( w ) define F = Flw ) We let can . . subgroup of ( isomorphic to ) Gull KI IE ) Claim a is hom ) ( Stntyg injective : create Gull Kele ) .
- rly . Deline it Gul ( KI IE ) Y ( r ) be given let - . element r fixes 41 r ) that ( Hlf ) ? in Since Is element have Hr ) fixes s E E , and F we since , Autfkf ) , element of 4G ) is show in F To an - Kf . that rlkf ) check - only to hue we . conjugate in only know its Galois kf/F is Since , we - Kf . r ( Kf ) itself Tcf is so - ,
check injecting , ,rzEGnl ( If .IE ) let be 's let r . - ra 414=464 , so tht r then given , . appropriate for . , an ) = Fla , a. → an , Kf know we . - - , an ) If - F ( w , ai , that so - . - rack ) for r , ( k ) - 14oz ) , Hr , ) Since know - - we - raki ) , Cai ) knew particular ktkf . - o we In my Gul ( KI IE ) fixes since w , . But all for Kien Gul ( Ef IE ) elements of - rzlw ) . Siwa r , ( w ) have - on generators of KI we , we get q=q . by Their action determined are
Gall KI IE ) subgroup at have is a we now : FIE ( Kf : F ) - N . [ KI Gul ( Kf/F ) Nate also . Gul CKIIE ) is , we get deep ish theorem By our { Hi ) ! . exist subgroups : there solvable = & E He He . , - Ho E Hz - e { c) H , E E - - - . Gullette ) Hit '/Hi and His Hit , cyclic is s . that .
on this chain Gaulois we'll the use correspencenie Now of subgroups qfcl-ttoa-H.EHE-i-EHe.IE/tega4q,p a Hit '/Hi and His Hit , cyclic s . that is . Ff - ← E " t - II. c- Ie . . ← I.e . . ← . - I ' . . . Gull " - Gul Kit . Iki ' 1M¥ ) Galois and and is
Gul ( Kit 't ; ) get Sinan cyclic , we is Kummer they - ki ( Vii ) from Tmt kit . - Hence kit ki have for we some . radical tower . E → If is a fact = FIKE ) F- = Flw ) But in s . , tower radical Fans If is a F ↳ Hence f solvable DMB Kf . conking is .
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