Drorbn at 20:11, 7 March 2007

2007-03-07T20:11:48Z

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{{07-401/Navigation}}

==The Test==

===Front Page===

<center>

'''Do not turn this page until instructed.'''

<font style="font-size:150%">Math 401 Polynomial Equations and Fields</font>

<font style="font-size:125%">Term Test</font>

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.

'''Notes.'''

</center>

* 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.

<center> '''Good Luck!''' </center>

===Questions Page===

'''Solve 5 of the following 6 problems.''' Each of the problems is worth 20 points. You have two hours. '''Neatness counts! Language counts!'''

'''Problem 1.'''
# 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").

'''Problem 2.'''
# 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 <math>{\mathbb Q}[x]/\langle x^2+1\rangle</math> is a field.

'''Problem 4.''' Let <math>R</math> be a commutative ring of prime characteristic <math>p</math>. Show that the ''Frobenius'' map <math>x\mapsto x^p</math> is a ring homomorphism from <math>R</math> to <math>R</math>.

(Remember that in math-talk the word "show" is equivalent to the word "prove").

'''Problem 5.'''
# Define "a principal ideal domain" (PID).
# Prove that if <math>F</math> is a field then <math>F[x]</math> 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").

<center> '''Good Luck!''' </center>

{{07-401/Results of the Term Test}}

==Solution Set==

Students are most welcome to post a solution set here.

07-401/Term Test - Revision history

Drorbn at 20:11, 7 March 2007