Unformatted text preview: (c) Prove that Z 10 has an idempotent other than 0 or 1, and hence has zero divisors. Solve for its last 4 digits. (Hint: Solve for the last digit, and then show that a version of Hensel lifting works in this case even though 10 is not prime.) (d) Prove that the zero divisors in Z 10 yield the elements in Z / ( 10 n ) that correspond, by the Chinese remainder theorem, to the elements ( 1 , ) and ( , 1 ) in Z / ( 5 n ) × Z / ( 2 n ) . GK2.3. Let p be a prime number. Prove that a padic integer in Z p has a reciprocal in Z p if and only if its units digit is nonzero. *GK2.4. Prove this structure theorem for the ring of nadic integers for any integer n : If n = p e 1 1 p e 2 2 ··· p e k k is the prime factorization of n , then Z n ∼ = Z p 1 × Z p 2 ×···× Z p k . (The exponents aren’t used.) 1...
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 Fall '08
 Shapiro,I
 Math, Number Theory, Prime number, 4 digits, 12 K, divisors.

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