::::::i function = O c- zero - - - t en Tn C , et , t = O = eye - PowerPoint PPT Presentation

Independence of charades of Characters Independence III : ( OPTIONAL )

In depone of of auratus : Part I From ÷÷÷÷:÷:÷÷÷÷÷:÷:÷:÷:÷÷÷i function = O c- zero - - - t en Tn C , et , t = O = eye Then e , = - - - . NewSt result Proving this

Det ( Character ) afield E character and G a For , group a s E # hem X : G - is G E a of . over , then - IR G - Glen ( IR ) let and EI E - let Glen ( IR ) → IR # character . is a dot :

Defy ( Independence of chants ) . . , Xr ) of I E is { Hi , over Chanute 's at A set satisfying values of . . , er E E only a , if the independent - er = 0 =D - - t er Xr - e , - - - - are e , X , - t , Cfn dependence of Charters ) c- Dedekind theorem characters of Gour E are of distinct collection finite independent Any . are given , . . ,rn3E Aut LE ) care ? If Er . . why do we . By E E # of rile # a character over then is - independent . " E ' ' They are indepuue of charters .

number of characters ) on The induction ( by PI . If chanter - O ex be X a - let ! Basics , - O ? Ng ) toe E Nate e - say we can eXCg ) to , then GEG . If for all . eto ① - O implies e - O - ex Hence . - - I of collection r Assume Inductive my independent : characters is . distinct or fewer independent each other ) where . . , Hr ) are { Hi , want We due tinct from Xi the are

that have So . - yer EE so e. , we suppose =D t en Xr e , X , t - - - . This means tg EG - ter Xrlg ) - O e. X. (g) - t - by induction eino , then if that ay observe First , all zero . are ei know we coefficients nonzero are ei The assume all . So : assume . ' can by through er , we By scaling tg EG - ter . . Xmlgltxrlg ) - O e. X. (g) - t ① - .

HEG X. Ch )tXrlh ) X , # Hr , we have so some Since . - { hg :gEG } becomes = G ,eq' n I HG Because - - ter . .Xr .lk/tXrlhg)--OV-gEG e. X. ( hg ) t ① - . - O tg - - ten . Xralhtxr . . (g) + Xrlhlhlrlgl . ⇒ e , X. lhtxilglt - ' Xrlh ) Multiplying through by : - ' High - - ter . . xrdhlxrlhtkr.ly/tXrlg)=ofg e. X. MARCH - ② and ② Subtract ① :

Fg EG - ter . . Xmlgltxrlg ) - O e. X , (g) - t ① - . - ' High - - ter . . xrdhlxrlhtkr.ly/tXrlg)=ofg e. A. MARCH - ② - - ' ) Hr , Ig ) =D - - ter . , ( l - Xrnlh )Xrlh5 " )Hlg ) t - x. lhtxrlh ) ell . Hg . . , Xm ) - combination of six . . E This is an inductive hypothesis , all equals O . By that " ) to . e , ( I - x. lhlxrlhl coefficients → a- 0 But are . Dna