Solutions to (Selected) HW Problems
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چکیده
We claim that the prime/irreducible elements of R are the associates of primes in Z different from p. Indeed, suppose that a/p, with a ∈ Z is an irreducible. Since powers of p are units, we may assume k = 0 and p a. Since −1 is also unit, we may also assume a > 0. If a is not prime, it factors in Z as a = bc, with b, c 6= ±1. Since p b, c [as p a and p is prime], they are not units of R and hence a is reducible in R. Now, if a is irreducible in Z, then it is a prime of Z different from p [as a > 0]. Suppose a = (b/p)(c/p). Then, ap = bc in Z, and since a is prime [in Z], we have a | b or a | c. But that being the case, by unique factorization [in Z], we have that either b/p or c/p is a power of p, i.e., a unit. Thus, a is irreducible.
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Solutions to (Selected) HW Problems
Please read: I will try to post here a few solutions (or answers). The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” w...
متن کاملSolutions to (Selected) HW Problems
Please read: I will try to post here a few solutions (or answers). The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” w...
متن کاملSolutions to (Selected) HW Problems
Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” were the same,...
متن کاملSolutions to (Selected) HW Problems
Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the “answer”. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the unexplained answers. Be careful that just because our “answers” were the same,...
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