(15 points) Let p be a prime and n E N. (a) Assume that a, b € Z are relatively prime and that pn | ab. Prove that p" |

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15 Points Let P Be A Prime And N E N A Assume That A B Z Are Relatively Prime And That Pn Ab Prove That P 1 (62.22 KiB) Viewed 60 times

(15 points) Let p be a prime and n E N. (a) Assume that a, b € Z are relatively prime and that pn | ab. Prove that p" | a or pn | b. (b) Prove that the only roots of X² – X in the ring R Z/pZ are OR and 1R. (Hint: Suppose that p=r+p"Z is a root. Use (a) to show that r = 0 (mod p") or r = 1 (mod p"). Keep in mind that, for n ≥ 2, R is not an integral domain.) =