Math 115B Homework 2 - (c Prove that Z 10 has an idempotent...

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Math 115b: Number theory Homework 2 This problem set is due Friday, January 25. GK2.1. The question arises of when i is present in Z / p for different primes p . In other words, when there is a solution to the equation x 2 = - ¯ 1, which you can then call i . (a) Call two elements a and b of Z / p friends if a = - b or a = ± ib . Prove that ¯ 0 is friends with itself, and otherwise that friends come in sets of four. (b) Corollary: i 6∈ Z / p if p 3 ( mod 4 ) . GK2.2. If two non-zero elements a and b of a (commutative) ring R (which may be the same element) satisfy the equation ab = 0, then they are called zero divisors . A ring without zero divisors is called an integral domain , and it is considered nicer than one that has zero divisors. In other words, in an integral domain, the product of any two non-zero elements is non-zero. (a) Prove that if p is prime, then the ring Z p is an integral domain. (b) If a 2 = a in a ring, then a is called an idempotent . Show that if a is an idempotent other than 0 or 1, then 1 - a is also an idempotent and they are both zero divisors.
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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 p-adic integer in Z p has a reciprocal in Z p if and only if its units digit is non-zero. *GK2.4. Prove this structure theorem for the ring of n-adic 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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