Number Theory

The Quadratic Reciprocity Law: A Collection of Classical by Oswald Baumgart

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By Oswald Baumgart

This booklet is the English translation of Baumgart’s thesis at the early proofs of the quadratic reciprocity legislation (“Über das quadratische Reciprocitätsgesetz. Eine vergleichende Darstellung der Beweise”), first released in 1885. it really is divided into elements. the 1st half offers a really short background of the improvement of quantity conception as much as Legendre, in addition to designated descriptions of numerous early proofs of the quadratic reciprocity legislation. the second one half highlights Baumgart’s comparisons of the rules in the back of those proofs. A present record of all identified proofs of the quadratic reciprocity legislations, with entire references, is supplied within the appendix.

This ebook will attract all readers drawn to uncomplicated quantity idea and the historical past of quantity theory.

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Extra resources for The Quadratic Reciprocity Law: A Collection of Classical Proofs

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Thus g < q 2 1 . 20) imply the following: assuming6 q > p, then the residue systems xq mod p and yp mod q contain all integers [less than] p 1, each half of them. The number of residues in yp mod q is q 2 1 , hence this system contains q p residues larger than p. If G 0 among them are even and U 0 are odd, then we 2 have U 0 C G0 D q p : 2 Moreover we easily see that GD p 1 2 C G0; U D p 1 2 C U 0; which implies U G Á U 0 C G0 Á q p 2 mod 2: 6 Proof by Zeller [73] 1. According to GAUSS’s Lemma we have .

The exponents of the p 1 terms inside the brackets are just the integers 1; 2; : : : ; p 1 since g is a primitive root modulo C1 p. Since the signs alternate, we see that x g xg ˙ : : : D ˙G. The sign of G is that of . 1/p x, and since p is odd we conclude that ˙G D . 1/ G. g 2 / Á . pq / mod p, and since g p 1 2 Á 1 mod p, this implies . 25) 2. x / D 1 C x C x g C : : : C x g . x / is, because g is a primitive root modulo p, divisible by 1 x p , hence by p 1 x p . x / will be divisible by 11 xx if 1 x 1 xp 1 xp Á 0 mod : 1 x 1 x For a proof we have to distinguish two cases.

Pq / mod p, and since g p 1 2 Á 1 mod p, this implies . 25) 2. x / D 1 C x C x g C : : : C x g . x / is, because g is a primitive root modulo p, divisible by 1 x p , hence by p 1 x p . x / will be divisible by 11 xx if 1 x 1 xp 1 xp Á 0 mod : 1 x 1 x For a proof we have to distinguish two cases. (I) a n d p a r e c o p r i m e. x / is divisible by 11 xx . (II) a n d p a r e n o t c o p r i m e. x / p is divisible by 11 xx . Collecting everything and recalling that g 0 C 1, g C 1, . . , g p the numbers 2, 3, .

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