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MAT-115A PROBLEM SET 9

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3 be a prime number. Show that x2??3(mod p) is solvable i p? 1(mod 6).5. Prove the following: if p = 4k + 1 and d k, then (d/p) = 1.6. Let p be a prime number. We call a unit a in Z/pZ a primitive root, if ordp (a) = p? 1, i.e. anyunit in Z/pZ can be written as some power of a. If p is of the form 2n + 1, prove that the primitive roots inZ/pZ are precisely the quadratic non-residues modulo p. If n > 1, prove 3 is always a primitive root.7. Let p be an odd prime, and (a, p) = 1. Show that if x2 = a(mod p) has solutions, then x2 = a(modpN) always has solutions, for any N > 1.8. Does x2 + x + 1? 0(mod 997) have solutions? Why or why not?

 

Paper#60653 | Written in 18-Jul-2015

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