II are : created ? Gal LEIF ) Lastine Gull Elf ) - f Ift F } Idf ) - PowerPoint PPT Presentation

The Galois Group elements How in II are : created ? Gal LEIF )

Lastine Gull Elf ) - f Ift F } Idf ) ={ rebut - iff - idf de re Gal ( EIF ) ie , - and th ) C- FIX ) has temma If a- Gal ( EIF ) LEE , Then f- ( rt ) ) too . - O f- (a) =D for - F - polynomials roots of permute ie , automorphisms

distinct rook and COI If ffx ) e F [ x ) has a Gal CE le ) s Sn field , then splitting its is E . on groundwork from & Rachel Gabby Building : field ) Galois group for splitting Corn ( order of FED , then a splitting field of Ttx . thisapp.lu?edthYfI:EtnQ If E is - FI - CE I Gal CEIFH ' -

iYYu Hmhg

" , an ) for algebraic lemme If - Fla , E - element re Gal CEH ) then elements an 4 , . . , an . , . action its a , . . , an determined on by . is . know the . . , ok . ) , values Ha , ) , if Ie , . we acts EEE how t know any on Then . we is given by for PI E - basis F An : oeeisdlirf-ca.ai.im ) ) . . - men { he '

Gabby extension of the ideas That ( This is an of The proof degree in The us gave . ) term . la element EEE means that can This every F - combination expressed uniquely an as be at basis this : " . . . . en die Hence a ?÷m , ' - - an te - e = . . .

we get r( e) =o ⇐ fq , " ) . . . . en die ' - - an te So - . . - men ) - e . he Eap . muts ' t ( te = - - , - - - Hagen . . - en ) da , )e ' rlfe Exponents = . " = Z - - r kn ) r la , )e ' te . . . - en . DM .

Gul ( QK , )/Q ) where Ey what is 4=352 ? on Hwy ( you saw this irala , )=x3 -2 Notation , - WZVZ roots w T2 other and : 43 - q= - t = J - It F3 w2= and 2- EQ where - w - " 3rd root of unity w 3=1 " since Note : is w . a

la ) That 6 92,234 show homework On you . run , )E{ a .az , as } . re Gall Okla ) 1a ) has Nate re Aut IQK , ) ) hand other know the we On , old , ) EQ ( a ) hence and . ok , ) Hence - a . - . , ai } Okla ) - basis for { 1,4 Q Nate is a

last result knew That and by we a our determined by at Gal ( ⑥ (a) 1a ) its is given action a . on . be idek rt Gal Cala ) 1a ) Huie must any . , - a , t q , -42 ) = r ( q , . It qz de ) = Hq . ) r IH tha ) r Kil tr ( q , ) ok . ) ' = q , ' - I tqz.at E3 'd , a e .

- Lida .ca , } Gal ( Okla ) 1a ) So - : . result that contradict doesn't our Nele : This result this : F ) = ( E I Gal ( El F) I because fields splitting only to applies . ④ (a) you not actually find that is a We field splitting .