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Proof let a be an integer relatively prime to p(not

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Unformatted text preview: Proof. Let a be an integer relatively prime to p (not necessarily odd), and let us adopt the same notation as in the proof of Theorem 9.5. Note that ja = p b ja/p c + α j , for 1 ≤ j ≤ k , so we have ( p- 1) / 2 X j =1 ja = ( p- 1) / 2 X j =1 p b ja/p c + n X j =1 r j + k X j =1 s j . Also, we saw in the proof of Theorem 9.5 that the integers s 1 ,...,s k ,p- r 1 ,...,p- p n are a re-ordering of 1 ,..., ( p- 1) / 2, and hence ( p- 1) / 2 X j =1 j = n X j =1 ( p- r j ) + k X j =1 s j = np- n X j =1 r j + k X j =1 s j . Subtracting, we get ( a- 1) ( p- 1) / 2 X j =1 j = p ( p- 1) / 2 X j =1 b ja/p c - n + 2 n X j =1 r j . Note that ( p- 1) / 2 X j =1 j = p 2- 1 8 , which implies ( a- 1) p 2- 1 8 ≡ ( p- 1) / 2 X j =1 b ja/p c - n (mod 2) . If a is odd,this implies n ≡ ( p- 1) / 2 X j =1 b ja/p c (mod 2) . 58 If a = 2, this — along with the fact that b 2 j/p c = 0 for 1 ≤ j ≤ ( p- 1) / 2 — implies n ≡ p 2- 1 8 (mod 2) . The theorem now follows from Theorem 9.5. 2 Note that this last theorem proves part (4) of Theorem 9.4. The next theorem proves part (5). Theorem 9.7 If p and q are distinct odd primes, then ( p | q )( q | p ) = (- 1) p- 1 2 q- 1 2 . Proof. Let S be the set of pairs of integers ( x,y ) with 1 ≤ x ≤ ( p- 1) / 2 and 1 ≤ y ≤ ( q- 1) / 2. Note that S contains no pair ( x,y ) with qx = py , so let us partition S into two subsets: S 1 contains all pairs ( x,y ) with qx > py , and S 2 contains all pairs ( x,y ) with qx < py . Note that ( x,y ) ∈ S 1 if and only if 1 ≤ x ≤ ( p- 1) / 2 and 1 ≤ y ≤ b qx/p c . So | S 1 | = ∑ ( p- 1) / 2 x =1 b qx/p c . Similarly, | S 2 | = ∑ ( q- 1) / 2 y =1 b py/q c . So we have p- 1 2 q- 1 2 = | S | = | S 1 | + | S 2 | = ( p- 1) / 2 X x =1 b qx/p c + ( q- 1) / 2 X y =1 b py/q c , and Theorem 9.6 implies ( p | q )( q | p ) = (- 1) p- 1 2 q- 1 2 . That proves the first statement of the theorem. The second statement follows immediately. 2 9.3 The Jacobi Symbol Let a,n be integers, where n is positive and odd, so that n = q 1 ··· q k , where the q i are odd primes, not necessarily distinct. Then the Jacobi symbol ( a | n ) is defined as ( a | n ) := ( a | q 1 ) ··· ( a | q k ) , where ( a | q j ) is the Legendre symbol. Note that ( a | 1) = 1 for all a ∈ Z . Thus, the Jacobi symbol essentially extends the domain of definition of the Legendre symbol. Note that ( a | n ) ∈ { , ± 1 } . Theorem 9.8 Let m,n be positive, odd integers, an let a,b be integers. Then 1. ( ab | n ) = ( a | n )( b | n ) ; 2. ( a | mn ) = ( a | m )( a | n ) ; 3. a ≡ b (mod n ) imples ( a | n ) = ( b | n ) ; 4. (- 1 | n ) = (- 1) ( n- 1) / 2 ; 5. (2 | n ) = (- 1) ( n 2- 1) / 8 ; 6. if gcd( m,n ) = 1 , then ( m | n )( n | m ) = (- 1) m- 1 2 n- 1 2 ....
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Proof Let a be an integer relatively prime to p(not...

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