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- May 5th 2008, 04:25 PM #1
## need some help on these

1) Let p be an odd prime. Prove that there is a primitive root of p

whose (p-1)th power is NOT congruent to 1 mod p^2.

Hint: Let g be any primitive root of p. If g works, done.

If g doesn't work, then show that the primitive root g+p of p

does work.

2) For n > 2, let M = 2^n. Prove by induction on n that

5^(M/8) is congruent to 1 + M/2 mod M.

3) Show that the solutions to x^e - 1 = 0 mod p in RRS(p)

are the d numbers b, b^2, ..., b^d mod p, where d=gcd(e,p-1)

and b is an element mod p of order d.

Hint: Write d as a linear combination...

- May 5th 2008, 05:25 PM #2

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- May 5th 2008, 05:30 PM #3

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- May 5th 2008, 08:50 PM #4

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Let be primitive root of (assuming it is odd). Say solves . Then and so this means . There are solutions for . This tells us that has at most solutions. It is easy to check that are all distinct and each solves . Using the above result that there are at most solutions it tells us that is a complete set of solutions (up to congruence).