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Unformatted text preview: N k ( q, h ) = k = ( h ) . 3. Suppose that k  ( p 1) and let be a character modulo p oF order k , say ( n ) = e ((ind n ) /k ). (a) Prove that For all h , N k ( p, h ) = 1 + k 1 j =1 ( h ) j . (b) Prove that iF p a , then G k ( p, a ) = k 1 j =1 ( a ) j ( j ) . (c) Prove that iF p a , then  G k ( p, a )  ( k 1) p and hence in general  G k ( p, a )  (( k, p 1) 1) p . 4. Let M s ( q, n ) denote the number oF solutions in x 1 , . . . , x s oF the congruence x k 1 + + x k s n (mod q ). (a) Prove that M s ( q, n ) = 1 q q a =1 G k ( q, a ) s e ( an/q ). (b) Prove that  M s ( p, n ) p s 1  (( k, p 1) 1) s p s/ 2 ....
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This note was uploaded on 01/23/2012 for the course MATH 571 taught by Professor Vaughan,r during the Spring '08 term at Pennsylvania State University, University Park.
 Spring '08
 Vaughan,R
 Number Theory, Congruence, Integers

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