Solutions to (Selected) HW Problems
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چکیده
1. Suppose that R is a UFD. Let a(x) = ca1(x) and b(x) = db1(x) be primitive decompositions of a and b. Then, since p(x) ∈ R[x], we have that cd ∈ R. Now, the leading coefficient of a(x)b(x) is the cd times the product of the leading coefficients of a1(x) ∈ R[x] and b1(x) ∈ R[x]. But since p is monic, all these must units of R. So, without loss of generality, we may assume that a1 and b1 are monic, and hence cd = 1. But both a(x) = ca1(x) and b(x) = db1(x) are monic [as well as a1 and b1], we have that c = d = 1, and hence a = a1 ∈ R[x] and b = b1 ∈ R[x]. So, if a 6∈ R[x], R cannot be a UFD.
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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...
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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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