I : Newstead definitions Basic mob of unity in a study as - PowerPoint PPT Presentation

Roots of Unity Basics of Roots of Unity I :

Newstead definitions ① Basic mob of unity in a ② study as subgroups of # ③ roots of unity F

roots of a field , then The Define If F is nth root of unity the called c- FIX ] - I " are X in F . nots of unity nth most Nate has at F n roots of unity relatively few EI has IR even " roots :{ I , - t ) - l ) { l , 2nd roots i . root of unity 113 odd th :

nots of ⇐ unity care about why do ? we ① Ipa " of splitting - x over Ip field XP - GF lpn ) is the - over Ip is the splitting field for " - t - I xp Gffpn ) - Dst collection of is just th Cpa = GF ( pm Epn so : , over Ip with O ) ( together root of unity by adjoining finite fields create Hence : we unity Kp of to roots

splitting field for studied Ex ) ② we've x ' - 2e also E) we've seen called it ( we extensively . root of , Then x3 -2 some that if is a However ⑨ K ) t E : . binomials ) ( Reds of unity and theorem , then if F C- FIX ) splits in xn - I Suppose that c- Flex ] , - f of " root contains x x one KIF K xn - f splits in get Then . we

roots of unity in a ' analyze Uts - set The Then ne IN let . : " " } = { cos 12¥ ) ti sm ( HI ) . . ask an } " %) " { ( e - I =D over Cl . solutions to dish not " of x set n is a root of unity in ¢ ) of set nth ( ie , this is the . ) problem in the set next ( this up comes

what degree behavior et mob of unity is the To a general field ? carried to over ¢ in roots of nth the a general field , are In # ? yes of ! F subgroup cyclic unity a F # are cyclic ) subgroups of Theorem ( Finite IGKA , then is cyclic IFGE F # G . and text ) is Theorem 62 in ( this

PI with - IGI sis We F # - GE n let . cyclic want : G is . of Finite Ab . Groups . Thm Fund the from Recall . - - ⑦ Ink = Km , ④ G have we . - IGI - - Mk . Observe m , - = : . n . - - ink ) { mi , are iff cyclic also : G Recall is Mk ) 1cm Em , , n pairwise relatively - - prime ⇐ . mis ) - - Mk slam { mi , we already know Note that - M , n - .

Then for Mk ) - 1cm { m , , write If - m we . corresponding get g EG we a any , - - - ④ Kink says c- Km Lagrange - y ga ) ④ ( g , , . , ⇒ lgillm all Igi l l mi for i . . . . , gal I heave and Ilg . . get m Hence , we GEG So all lgllm for Hence - I gm - . . NHGIEM Hence roots of XM - I g. EG all 1¥ are , .

, then a field and NEIN tar If F is cyclic F term unity roots of in a nth the of F # subgroup .