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Math 401 Polynomial Equations and Fields
University of Toronto, February 28, 2007
Solve 5 of the 6 problems on the other side of this page.
Each of the problems is worth 20 points.
You have two hours to write this test.
- No outside material other than stationary and a basic calculator is allowed.
- Please stay around when you are done writing. Following the test and following a short break, we will have some further discussion in the examination classroom.
- The final exam date was posted by the faculty - it will take place on the evening of Tuesday April 24 between 7PM and 10PM, at New College Residence (NR) room 25.
- Neatness counts! Language counts! The ideal written solution to a problem looks like a proof from the textbook; neat and clean and made of complete and grammatical sentences. Definitely phrases like "there exists" or "for every" cannot be skipped. Lectures are mostly made of spoken words, and so the blackboard part of proofs given
during lectures often omits or shortens key phrases. The ideal written solution to a problem does not do that.
Solve 5 of the following 6 problems. Each of the problems is worth 20 points. You have two hours. Neatness counts! Language counts!
- Give an example of a finite noncommutative ring.
- Give an example of an infinite noncommutative ring that does not have a unity.
(Your examples must be clearly stated and you must provide a few words of explanation why your examples "do the right thing").
- Define "an integral domain".
- Define "a field".
- Prove: A finite integral domain is a field.
(As always in math exams, when proving a theorem you may freely assume anything that preceded it but you may not assume anything that followed it).
Problem 3. Prove that the quotient ring is a field.
Problem 4. Let R be a commutative ring of prime characteristic p. Show that the Frobenius map is a ring homomorphism from R to R.
(Remember that in math-talk the word "show" is equivalent to the word "prove").
- Define "a principal ideal domain" (PID).
- Prove that if F is a field then F[x] is a PID.
Problem 6. Construct a field of order 25.
(Your construction must be clearly explained and you must provide a few words of explanation why your construction "does the right thing").
A total of 35 students took the exam; after some appeals the average grade is 64.49 and the standard deviation is 27.62.
The full list of grades is: (11 19 28 28 30 31 32 33 37 48 48 50 50 57 61 62 62 64 66 68 72 79 80 81 83 96 96 96 97 97 98 98 99 100 100).
The results are quite similar to what I expected them to be, though a bit on the low side.
How should you read your grade?
- If you got 100 you should pat yourself on your shoulder and feel good.
- If you got something like 95, you're doing great. You made a few relatively minor mistakes; find out what they are and try to avoid them next time.
- If you got something like 80 you're doing fine but you did miss something significant, probably more than just a minor thing. Figure out what it was and make a plan to fix the problem for next time.
- If you got something like 60 you should be concerned. You are still in position to improve greatly and get an excellent grade at the end, but what you missed is quite significant and you are at the risk of finding yourself far behind. You must analyze what happened - perhaps it was a minor mishap, but more likely you misunderstood something major or something major is missing in your background. Find out what it is and try to come up with a realistic strategy to overcome the difficulty!
- If you got something like 35, most likely you are not gaining much from this class and you should consider dropping it, unless you are convinced that you fully understand the cause of your difficulty (you were very sick, you really couldn't study at all for the two weeks before the exam because of some unusual circumstances, something like that) and you feel confident you have a fix for next time. The deadline for dropping a class this semester is soon: Sunday March 11.
Remember! Grading is a difficult process and mistakes always happen - solutions get misread, parts are forgotten, grades are not added up correctly. You must read your exam and make sure that you understand how it was graded. If you disagree with anything, don't hesitate to complain! Dror graded everything, so appeals should go directly to him.
The deadline to start the appeal process is Wednesday March 21 at class time.
Students are most welcome to post a solution set here.