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Unformatted text preview: HOMEWORK PROBLEMS FOR MATH 8320: 29-47 PETE L. CLARK Here are some homework problems for the 8320 course. Some of them are repeated from places in the lecture notes. Some of them are not specifically tied to any one topic. Some of them are easy, and some are quite difficult, etc. 1 There are more problems than any one student will want to work out, but every student should work at least some of them and be prepared to present solutions to some of them. Unless mention is made to the contrary, k denotes an arbitrary field, but you have my permission to assume that k is perfect if you dont like inseparable field extensions. 2 If an exercise is labelled with an L, that means that substantial help on that exercise can be found in Lius book. (The converse does not necessarily hold...) Exercise 29: Determine which PIDs have the Hilbert-Jacobson condition. Then (if you know a bit about Dedekind domains), conclude that a Dedekind domain is either a Hilbert-Jacobson ring or a PID (or both). Exercise 30: For an affine k-algebra A, show that the following are equivalent: (i) A is finitely generated as a k-module. (ii) A has dimension zero. (iii) The set MaxSpec( A ) of maximal ideals of A is finite. Exercise 31: Classify all affine k-algebras A for which every ideal is a radical ideal. Exercise 32L: Suppose that k is a field which is separably closed (but not nec- essarily algebraically closed), and let V be a geometrically integral affine k-variety. Show that V necessarily has a k-rational point, i.e., a maximal ideal whose residue field is just k itself. (Hint: use part b) of the Noether normalization theoem.) Comment: A field k is called pseudoalgebraically closed (PAC) if every geomet- rically integral k-variety has a k-rational point. Thus this exercise demonstrates that any separably closed field is PAC. There are, however, many other PAC fields,that any separably closed field is PAC....
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