Let p be a prime and n E N. a) Assume that a, b = Z are relatively prime and that p" | ab. Prove that p" | a or pr | b.

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Let P Be A Prime And N E N A Assume That A B Z Are Relatively Prime And That P Ab Prove That P A Or Pr B 1 (18.07 KiB) Viewed 52 times

Let p be a prime and n E N. a) Assume that a, b = Z are relatively prime and that p" | ab. Prove that p" | a or pr | b. b) Prove that the only roots of X2-X in the ring R = Z/p"Z are OR and 1R. (Hint: Suppose that p=r+pZ 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.)