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Roots of Unity & Quadratic residues

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  • 2. <%'='&'>* 1&: %..& .0 "1'&/ ?0 +* @7 A '+ #1 1&: %..&.0 "1'&/ '* 9 +*1$ #1$ '0 BC7 4 0.% #44 =*1 &:*1 :* '+ D1.E1 #+ # -%'='&'>* 1&: %..& .0 "1'&/;
  • 3. ?0 " '+ # -%'='&'>* 1&:+..& &:*1 ? F F 1.9 .$9 .D 9 ; ; ; +*1& #%*#44 1&:%..&+ .0 "1'&/ 9 +'1(* G"'H I JK'#4 #1$ &:* *&&+ 1. 9 1$ 9 .0 ; ; F +*1& #1 #44 $'+&'1(& +'1(* 1.#1$ L +(' F '& E:*%* 'F 0#1E:'(: '+ # (.1&%#$'(&'.1;
  • 4. B. .0 1&: %..&+ .0 "1'&/ '1 # 0'1'&* 0'*4$; <%.-.+'&'.1 2 :*& 8 M* # 8*1*%#&.% .0 780; N:*1 8$ '+ #1 1&: %..0 .0 "1'&/ '00 18 5 . =.$ J F K N:* 1"=M*% .0 1&:%..&+ .0 "1'&/ '+ #8*$ O 1 9 ,; F K ;
  • 5. <%..02 6*& %8 M* # 8*1*%#&.% .0 7AP; N:*1 *>*%/ *4*=*1& '+ .0 &:* 0.%= 8Q B.E 9 8'& '00 A F '0R O +'1(* S(8TF UFK '; G8'HV ? '00 8 1$ #4 '00 ,W ?1R '00 1R 5 . =.$ X F K ;
  • 6. B.E 4*& $;5 8*$(19 8FK;L O1# 9 Y L N:*1 8''+ #1 1&: %..& .0 "1'&/ '00 ZR @. =.$*4; F K '00 1# R* . =.$ G$&H '00 R* . =.$ G:#&H +'81;88[$9,%/$HF/ '00 R '+ # ="4&'-4* .0 G:#&H
  • 7. '* 9 8''+ #1 1&:%..& .0 "1'&/ 0.% #44 R 5 ="4&'-4*+ .0 #$$ N:*/ #%* 8$9 8#&*$H 9 8$&&$ H 9 ; ; ; 9 8$ N$$H 5[ N:"+ &:*%* #%* $ $'+&'1(& -.E*%+ .0 8 ?$ O'0 &:*/ E*%* 1.& $'+&'1(& 9 8 ] ^K4$ 5 9 E:*%* G'F RH _K %# (.1&%#$'(&'.1 Y`F ?
  • 8. N:*/ #%* #44 1&: %..0+ .0 "1'&/ ; a. 1"=M*% .0 1&: %..&+ .0 "1'&/ 5 $ 5 8.$G1 9 8 F K
  • 9. N:"+ &:*%* #%* 1 +"(: %..&+ '00$51; '; @A Y& :#+ -%'='&'>* 1&:%..& .0"1'&/ .00 $FF 1 '00 1 $'18*$#&9K ?1 &:'+ (#+* 9 '* ; 9 '0 $51 9 &:*1 *>*%/ 1&: %..& .0 "1'&/ E'44 M* .0 &:* 0.%= 8' E:*%* R F ="4&'-4* .0 `?B
  • 10. 4*& 8 F F 8b F KC 8'(-D1&c0;`` F K'18'+ 8' N:*1 *>*%/ 1&:%..& .0 "1'&/ E'44 M* .0 &:* F 0.%= 8'; d.E '+ 8' # -%'='&'>* +..& I 8'>* 9 '0 J ^F C8=& eK .#0'+: ;4D[K'00J;FK'?15# ="4&'-4* .0`FK fL@1$ '00 RD# '+ #1 '1&*8*% '00 RD 5 . =.$ 1 ;
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