":::i too big ) aren't CI ( Galois groups [ E : F) 31gal ( - PowerPoint PPT Presentation

Galois Extensions " big " ? Galois group ' when I : are

" Independence of " Chohuters , Part I "÷÷÷÷÷÷:÷÷::÷÷÷i÷÷ too big ) aren't CI ( Galois groups [ E : F) 31gal ( EIHL i F) sis . Then ( E Suppose ( Gal ( El F) I > [ E Gal ( E " " ) : E z . k ④ " iff Ga ' CE 't ) = f I Gal LEIF ) l " E Note : is size Max

Extension ) or Normal n ( Galois Extension ' Def Galois extension called AT is a Elf extension - l Gal LEIF ) l [ E : F ] if - . EGAKEIF ) iff f Galois is . EIF Nate extensions Galois characterize theme : Today 's the is already know that if E th ) c- FIX ? Nate we a superable polynomial for field splitting [ E : F) =L Gal LEIF )l Then .

extensions ) characterizations for Galois Thin ( Equivalent - Gal CE tf ) . Then [ E : F) co and let G - suppose equivalent following : are the - EG Cis F root in E , p G) C- FIX ) with some irreducible (2) for all plx ) splits in E is separable and plx ) have we some gamble fide ECD splitting field for is the E Galois (4) Elf is

⇒ of square of happiness City " " y ( we'll PI the use ④ 4) ⑤ 13 ) - EG get to ( CH ⇒ L2 ) ) F- assume we . some neat in and has irreducible : if is plx ) EF WTS in E . splits and separable pk ) is , then E a- E satisfying plant - O . with irreducible be plxteflx ) let WLOG that manic , plx ) is assume We can a irrp Ca ) . plx ) so

- sik ) : re Gal ( EIF ) ) = { a , - { da ) let 0 . - - no repeats - under of orbit a Gal LEIF ) = II ( x E E ( x ) - ri ) consider glx ) . Claim : glx ) @ FIX ) . Then given be c- Gal LLEIF ) . * gun ) ? # ( II. Ix let - II. Ix IIK-ait.gl#epobyrrke - ai ) ) - - - -

E Gall E' F) have Since can F , we = - glx ) c- Gal LEIF ) * ( glx ) ) all fer T T - come tram F coefficients of glx ) all iff the glx ) EF Cx ) iiff . - ai ) EFG ] - ITH polynomial glx ) - . have Upshot we : root of glx ) So . is a x : = ( irrr.CN/--lplxD . - O ) : hk ) { hlx ) c- FED - know We . . , Ga ) roots of plxl are from { a , So : plxllglx ) Hence . .

roots repeated and glx ) has So no since - a :) , glx ) : IT ( x " sub product of " plx ) is a roots repeated plxl has have no . we separable plx ) is . So : does plx ) splits in E , so glx ) . Also : since ⑤ irreducible plx ) c- FIX ) ( Lii ) ⇒ Ciii ) ) we get to any use : is separable splits and in E E with root . in a a separable polynomial splitting field of is The . E WTS : from FAT .

for some algebraic - pan ) - Fla , know E - We . E that claim - - in EE We X. , for II. irrplai ) . splitting field is the . irrflai ) separable ? By (2) since why is this , root ai has the we irreducible E and , is of By def 'n separable irrp ki ) is . know we get - irreducible polynomials separable for , non Rest of proton homework separable II. irrp ( ai ) is .