- Let P Be A Prime And N Is In The Set Of Natural Numbers 1 (241.83 KiB) Viewed 40 times
let p be a prime and n is in the set of natural numbers
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let p be a prime and n is in the set of natural numbers
let p be a prime and n is in the set of natural numbers
(a) Assume that a, b € Z are relatively prime and that p" | ab. Prove that p" | a or pn | 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.)
(a) Assume that a, b € Z are relatively prime and that p" | ab. Prove that p" | a or pn | 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.)