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Question: the roots of 1 in characteristic p for p prime...

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The th roots of 1 in characteristic p for p prime, nin mathbb{Z}_{+} . To consider x^{n} -1 in mathbb{Z}_{p}[x] .

a) From now on assume gcd(n,p)-1. Let K be the spitting field of r1 . Let R_{n} be the set of th roots of 1.  R_{n} is an abelian group, in fact, that R_{n}cong mathbb{Z}_{n} and that 2) for kequiv [K:mathbb{Z}_{p}] . Determine k in terms of p and n .

b) Use your answer to (a) to find k when 5 and n=12 , and when p=7 and 15 rl .

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