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quad-reciprocity - Gausss Quadratic Reciprocity Theorem...

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Gauss’s Quadratic Reciprocity Theorem : NumThy Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 30 November, 2011 (at 01:30 ) 1 : Nomenclature. For odd D , use H D to mean D - 1 2 . ( The H is to suggest “Half”. ) In the sequel, p is an odd prime and S p is the stridelength ; we will walk around the circumference= p circle using strides of length S . Use H := H p and hh x ii := hh x ii p for the symmetric residue of integer x modulo p ; so hh x ii is in [ H .. H ] . Let mean p . Let G = G p ( S ) be the set of indices [ 1 .. H ] such that hh · S ii p is neGative. Letting P be the indices with hh · S ii Positive, we have that ( disjointly ) G t P = [ 1 .. H ] . Finally, use N = N p ( S ) for the number of “negative” indices; N := # G . 2 : Prop’n. Fix an S p , with notation from (1) . Then the mapping ( absolute-value of symm-residue ) 7→ hh · S ii , is a permutation of [ 1 .. H ] . We say that the map- ping 7→ hh · S ii is a permutation up to sign of [ 1 .. H ] . Proof. Given indices 1 6 6 k 6 H , we want that either equality ∓hh · S ii = hh k · S ii forces = k . For either choice of sign in , note that ∓hh · S ii = hh k · S ii IFF 0 [ k ± ] · S IFF 0 k ± ‘ , since S p . Thus 0 6 k ± 6 2 H < p . Together with k ± 0, this forces k ± to actually be zero. Thus the “ ± ” is a minus sign, and k = . 3 : Gauss Lemma. Fix an odd prime p and integer S p . Then the Legendre symbol ( S p ) satisfies ( S p ) = [ 1] N . Proof of Gauss Lemma. Necessarily H Y =1 hh · S ii ≡ H Y =1 · S = H ! · S H H ! · S p ! , 4 : with the last step following from LSThm. Observe that hh · S ii equals ± hh ‘S ii as is-not/is in G . Prop’n 2, consequently, tells us that LhS(4) can be written as H ! times [ 1] N . Thus RhS(4) equals H ! · ( S p ) H ! · [ 1] N . The H !, being co-prime to p , cancels mod- p to hand us congruence ( S p ) [ 1] N . An important application is the following. 5 : Two-is-QR Lemma. Consider an oddprime p . Then 2 is a p -QR IFF p 8 ± 1 . Abbrev. An odd integer D is 8Near if D 8 ± 1; it is 8Far if D 8 ± 3. ( The names come from being, mod 8, near/far from zero.
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