I : Algebra : prove goal First Theorem of Algebra ) them ( - PowerPoint PPT Presentation

Applications theorem of Fundamental I : Algebra

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lemmas few preliminary prove this need To . , we a splits completely quadratic plxteclx ) Unmeet Any . roots of must know p PI if plx ) = axztbxtc , we ¢ ? - betrayal form Are these roots take in a- . 8 with contain to : does some E down This comes = xtiy ← rectangular cords If 82=62-4 ac b ' - Yao - rreik.com#eiaem.i*o=arctuPlxl = reit ← polar cord . so , r = rFty2 Consider r -

extension of degree 2 Core . has G no Hincks )=2 . - Ela ) where - 2 , K Pf If then Ck :& ) - - E is irreducible quadratic new That . no knew But we DADA root ) have ! real odd degree polynomials ( Real tenma has , Then f self ) - 2kt I with flx ) EIR Cx ) If - IR root on a . - linear irreducible . odd degree IRN has Cos , non no

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Algebra ) Theorem at Pf ( Fundamental plx ) HRW that WLOG - ① argue assume we can structure : Elk of ith ) pk ) . the splitting field ② consider ( Nde REQ - Rli ) E E ) . : IR ) =L Galois they [ E to argue Use in Cl plxl splits force will - E . Q This , so -

⇒ be assumed Rex ] be to in pH ① we can - { prove . fled =p ( x ) fl x ) it plx ) " an x , where - let - conjugate of an have pix ) ⇐ E. oaixn , where In - we = I * ( pm ) ) conjugation , then Cx ) is complex ( if I f iff flxtelklx ) flx ) ERIN ? know we why is T * ( fl x ) ) . But - ffx ) = f ( x ) - ftp.p-MHPH-flxl - ( E Han ) xn ) ( Etta ) xn ) * ( HH ) - T * ( plxtplx ) ) - . I

do knew root G , has in we Now : if f some Know : if fla ) G ? - O root in - has some , p pk ) plat - O Then - . complex seat . ( yay ! ) has a If Then p plat - O - , " =D Apply z : - O , then I End If Fla ) . - = phi ) " = E an E - Ethan ) In - TCO ) O - ( yay ! ) at root So a- EQ p is a .

let be th plxl.cl/2lx3 and E For ④ assume , IREQEE 1 × 2+1 ) pix ) This forces field splitting for . . - O ) , we ( since char ( E ) separable Iti ) pH - is Since - Gul LEHR ) let G Galois EHR - get . is . 2mk=lGal LEARN : IRI formula CE have degree By the we odd K and m > I is . for some ( Sylow theorem ) get that Some fancy group Theory we , pg lit Htt - 2M with HE Gulf EHR ) exists there . have [ FKH ) : R ) - R . Fundamental Theorem of Galois , - By we

of odd extension IR degree So FLA ) is . an LEECH ) VR FLH ) t IR let > I , then If K . . : 11231 [ FIH ) : IRI - k get - [ IRK ) , we d( irrp.la ) ) paper and is a so dlirr .pk ) ) of is ie an divisor k . , bigger than number 1 odd → ← ( to Lemmer 2) . heme and IGWILEHRH - 2 " k - - I , So . m=l but already want knew m3l , we We .

I Gall Ele ) I - 2M " , and m > I , Then [ am , If more fry group they says some ¢ HE Gall Ela ) 12 with exists there R Galois correspondence says - 2M£ we Htt - - lGYYI=Y÷ . - 2 [ Flit ) : e ) have - - . 2 extensions ! degree G has know But no we → ← : 1123=2 IREEEE Since CE , and - I Hence . , - m E - G . LE : =L formula gives se degree the ppg ,