16 The Quadratic Reciprocity Law - Caltech Mathematics Department

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$MATH 5b; Solutions to Homework Set 1 January 2009 1. As K is a ...$

Proof#1oftheorem: p, q odd primes, p = q. Put ξ = e 2πi q ∈ C <strong>The</strong>n ξ q =1, but ξ m =1ifm
This allows us to do “congruence arithmetic” mod p in R. ), Gauss introduced the following “Gauss Sum”: To study ( q p Sq = a mod q a ξ q a . Clearly, Sq ∈ R. Aside (Not part of proof of Quad. Recip., but interesting) So if Sq = 2 q−1 a=1 a q −1 =1,Sq = q ξ a + 2 q−1 a=1 −a ξ q −a ( −1 q )( a q ) ¯ ξa a (ξ q a + ¯ ξa ) ∈ R ∩ R and if −1 = −1, Sq∈ R ∩ iR q pure read or im. Lemma 1: Proof of Lemma 1: S 2 q = a mod q c mod q S 2 q =(−1) q−1 2 q a ξ q a a mod q b mod q b ξ q b = a b ξ q q a mod q b mod q a ξ b = ab ξ q a+b = ξ c a(c − a) q 3
Page 1: 16 The Quadratic Reciprocity Law Fi
Page 5 and 6: We proved earlier that so when c
Page 7 and 8: This is justified because ⇒ S p
Page 9 and 10: (This is part of a homework problem
Page 11: By Gauss’ lemma, q = p 2πqs

16 The Quadratic Reciprocity Law - Caltech Mathematics Department

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