Tout at 01:40, 5 April 2008

2008-04-05T01:40:07Z

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	==Solution Set==		==Solution Set==

			Solutions by Tout can be found [http://individual.utoronto.ca/tout/401TermTestSolution.pdf here]
	Students are most welcome to post a solution set here.

Drorbn at 20:05, 5 March 2008

2008-03-05T20:05:32Z

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			{{08-401/Navigation}}
	==The Test==		==The Test==
	===Front Page===		===Front Page===

Drorbn at 17:57, 2 March 2008

2008-03-02T17:57:03Z

New page

==The Test==
===Front Page===
<center>

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

Math 401 Polynomial Equations and Fields

Term Test

University of Toronto, February 27, 2008

'''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.
* The final exam date was posted by the faculty - it will take place on the ''evening'' of Tuesday April 28 between 7PM and 10PM, at BN2S (Large Gymnasium, South End, Benson Building, 320 Huron Street (south of Harbord Street), Second Floor).
* '''Neatness counts! Language counts!''' The ''\em 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.
* '''New!''' You will get 20% of the credit for any problem for which you will write explicitly "I don't know how to solve this problem" (whole problems only!).

<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.''' Consider the ring <math>\{0,2,4,6,8\}</math> under addition and multiplication modulo 10. Does it have a unity?

'''Tip.''' This, of course, is not just a yes/no question. You are expected to fully justify your answer, whatever it is.

'''Problem 2.'''
# Define "an integral domain".
# Define "a field".
# Prove: A finite integral domain is a field.

'''Tip.''' 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.''' Let <math>R</math> be a commutative ring with unity and let <math>A</math> be an ideal in <math>R</math>. Prove that <math>R/A</math> is a field if and only if <math>A</math> is maximal.

'''Tip.''' Don't forget! There are two directions to prove here!

'''Problem 4.''' Find all ring homomorphisms from <math>{\mathbb Z}/6</math> to <math>{\mathbb Z}/10</math>.

'''Tip.''' Here, of course, you have to explain both why the homomorphisms you found really are homomorphisms and why there are no more.

'''Problem 5.'''
# Let <math>F</math> be a field, let <math>f</math> be a polynomial in <math>F[x]</math>, and let <math>a</math> and <math>b</math> be two different elements of <math>F</math>. Prove that the remainder for the division of <math>f</math> by <math>(x-a)(x-b)</math> is <math>\frac{f(b)-f(a)}{b-a}(x-a)+f(a)</math>.
# Compute the remainder for the division exercise <math>\frac{x^{2008}}{x^2-1}</math>, done in <math>{\mathbb Q}[x]</math>.

'''Problem 6.''' Prove that the polynomial <math>f=3x^5+15x^4-20x^3+10x+20</math> is irreducible over <math>{\mathbb Q}</math>. If you are using any major theorem, you need to quote it in full, but you don't need to prove it.

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

{{08-401/Results of the Term Test}}

==Solution Set==

Students are most welcome to post a solution set here.

08-401/Term Test - Revision history

Tout at 01:40, 5 April 2008

Drorbn at 20:05, 5 March 2008

Drorbn at 17:57, 2 March 2008