(Problem 94: fixed latex)
 
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= Student solutions for Assignment #3  =
 
= Student solutions for Assignment #3  =
[[Solution_sample|Solution Sample]]  
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[[Solution sample|Solution Sample]]  
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== Problem 50  ==
 
== Problem 50  ==
*[[Media:Prob50.pdf|Problem 50 - Tan Dang]]  
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*[[Media:Prob50.pdf|Problem 50 - Tan Dang]]
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== Problem 73  ==
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Show that if <span class="texhtml">''p''</span> is a prime such that there is an integer <span class="texhtml">''b''</span> with <span class="texhtml">''p'' = ''b''<sup>2</sup> + 4</span>, then <math>\mathbb{Z}[\sqrt{p}]</math> is not a unique factorization domain.
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*[[Media:Problem_73_Zeller.pdf|Solution by Andrew Zeller]]
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::Here's my alternate proof - I found a few things that need to be changed in Avi's, which I can discuss in class.
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*[[Media:Prob_73.pdf|Solution by Avi Steiner]]
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::My solution only uses the fact that ''p'' is a sum of two squares (i.e. is congruent to 1 mod 4), so I'm not sure it's correct. -- Avi 20:05, 25 June 2013 (UTC)
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*[[MA553QualStudyAssignment3Problem73Solution|Solution by Ryan Spitler]]
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::I think this is a bit cleaner.
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::: I very much prefer this solution! -- Avi 11:08, 26 June 2013 (UTC)
  
 
== Problem 94  ==
 
== Problem 94  ==
Show <math>f(x) = x^4 + 5x^2 + 3x + 2</math> is irreducible over the field of rational numbers.
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*[[Media:Problem_94_-_Nicole_Rutt.pdf| Solution by Nicole_Rutt]]
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Show <span class="texhtml">''f''(''x'') = ''x''<sup>4</sup> + 5''x''<sup>2</sup> + 3''x'' + 2</span> is irreducible over the field of rational numbers.  
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*[[Media:Problem_94_-_Nicole_Rutt.pdf|Solution by Nicole_Rutt]]
  
 
== Problem 101  ==
 
== Problem 101  ==
(a) Show that x4 +x3 +x2 +x+1 is irreducible in Z3[x].
 
  
(b) Show that x4 + 1 is not irreducible in Z3[x].
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(a) Show that <span class="texhtml">''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1</span> is irreducible in <math>\mathbb{Z}_3[x]</math>.  
  
*[[Media:Week_3_Problem_101.pdf| Solution]]
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(b) Show that <span class="texhtml">''x''<sup>4</sup> + 1</span> is not irreducible in <math>\mathbb{Z}_3[x]</math>
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*[[Media:Week_3_Problem_101.pdf|Solution]]
  
 
== Problem 107  ==
 
== Problem 107  ==
Let <math>R</math> be a commutative ring with identity such that the identity map is the only ring automorphism of <math>R</math>. Prove that the set <math>N</math> of all nilpotent elements of <math>R</math> is an ideal of <math>R</math>
 
  
*[[Assn3Prob107]]  
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Let <span class="texhtml">''R''</span> be a commutative ring with identity such that the identity map is the only ring automorphism of <span class="texhtml">''R''</span>. Prove that the set <span class="texhtml">''N''</span> of all nilpotent elements of <span class="texhtml">''R''</span> is an ideal of <span class="texhtml">''R''</span>
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*[[Assn3Prob107|Solution by Avi Steiner]]
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== Problem 114  ==
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A local ring is a commutative ring with 1 that has a unique maximal ideal. Show that a ring R is local if and only if the set of non-units in R is an ideal.
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*[https://kiwi.ecn.purdue.edu/rhea/images/c/ca/Problem_114_Kent_State_Algebra_Qual_Ring.pdf Solution by Nathan Moses]<br>
  
 
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Latest revision as of 07:11, 26 June 2013


Student solutions for Assignment #3

Solution Sample


Problem 50

Problem 73

Show that if p is a prime such that there is an integer b with p = b2 + 4, then $ \mathbb{Z}[\sqrt{p}] $ is not a unique factorization domain.

Here's my alternate proof - I found a few things that need to be changed in Avi's, which I can discuss in class.
My solution only uses the fact that p is a sum of two squares (i.e. is congruent to 1 mod 4), so I'm not sure it's correct. -- Avi 20:05, 25 June 2013 (UTC)
I think this is a bit cleaner.
I very much prefer this solution! -- Avi 11:08, 26 June 2013 (UTC)

Problem 94

Show f(x) = x4 + 5x2 + 3x + 2 is irreducible over the field of rational numbers.

Problem 101

(a) Show that x4 + x3 + x2 + x + 1 is irreducible in $ \mathbb{Z}_3[x] $.

(b) Show that x4 + 1 is not irreducible in $ \mathbb{Z}_3[x] $

Problem 107

Let R be a commutative ring with identity such that the identity map is the only ring automorphism of R. Prove that the set N of all nilpotent elements of R is an ideal of R

Problem 114

A local ring is a commutative ring with 1 that has a unique maximal ideal. Show that a ring R is local if and only if the set of non-units in R is an ideal.


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