http://drorbn.net/index.php?title=11-1100/Homework_Assignment_4&feed=atom&action=history11-1100/Homework Assignment 4 - Revision history2024-03-29T02:10:51ZRevision history for this page on the wikiMediaWiki 1.21.1http://drorbn.net/index.php?title=11-1100/Homework_Assignment_4&diff=11061&oldid=prevDrorbn at 14:16, 17 November 20112011-11-17T14:16:33Z<p></p>
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</table>Drorbnhttp://drorbn.net/index.php?title=11-1100/Homework_Assignment_4&diff=11051&oldid=prevDrorbn at 14:06, 15 November 20112011-11-15T14:06:01Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 5.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 5.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.</div></td></tr>
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<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">----</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 6.''' (<ins class="diffchange diffchange-inline">Hard!</ins>) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.</div></td></tr>
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</table>Drorbnhttp://drorbn.net/index.php?title=11-1100/Homework_Assignment_4&diff=11050&oldid=prevDrorbn at 00:08, 15 November 20112011-11-15T00:08:03Z<p></p>
<p><b>New page</b></p><div>{{In Preparation}}<br />
{{11-1100/Navigation}}<br />
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This assignment is due at class time on Tuesday, November 29, 2011.<br />
<br />
===Solve the following questions===<br />
<br />
'''Problem 1.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>.<br />
<br />
'''Problem 2.''' (Selick) In a ring <math>R</math>, and element <math>x</math> is called "nilpotent" if for some positive <math>n</math>, <math>x^n=0</math>. Let <math>\eta(R)</math> be the set of all nilpotent elements of <math>R</math>.<br />
# Prove that if <math>R</math> is commutative then <math>\eta(R)</math> is an ideal.<br />
# Give an example of a non-commutative ring <math>R</math> in which <math>\eta(R)</math> is not an ideal.<br />
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'''Problem 3.''' (comprehensive exam, 2009) Let <math>A</math> be a commutative ring. Show that a polynomial <math>f\in A[x]</math> is invertible in <math>A[x]</math> iff its constant term is invertible in <math>A</math> and the rest of its coefficients are nilpotent.<br />
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'''Problem 4.''' (Lang) Show that the ring <math>{\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C}</math> is a PID and hence a UFD. What are the units of that ring?<br />
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'''Problem 5.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.<br />
<br />
----<br />
<br />
'''Problem 6.''' (Withdrawn, do not submit) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.</div>Drorbn