Difference between revisions of "11-1100/Homework Assignment 4"
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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. | '''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. | ||
− | + | '''Problem 6.''' (Hard!) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD. | |
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− | '''Problem 6.''' ( | + |
Revision as of 10:06, 15 November 2011
The information below is preliminary and cannot be trusted! (v)
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This assignment is due at class time on Tuesday, November 29, 2011.
Solve the following questions
Problem 1. Prove that a ring is a PID iff it is a UFD in which for every non-zero .
Problem 2. (Selick) In a ring , and element is called "nilpotent" if for some positive , . Let be the set of all nilpotent elements of .
- Prove that if is commutative then is an ideal.
- Give an example of a non-commutative ring in which is not an ideal.
Problem 3. (comprehensive exam, 2009) Let be a commutative ring. Show that a polynomial is invertible in iff its constant term is invertible in and the rest of its coefficients are nilpotent.
Problem 4. (Lang) Show that the ring is a PID and hence a UFD. What are the units of that ring?
Problem 5. (Dummit and Foote) In , find the greatest common divisor of and , and express it as a linear combination of these two elements.
Problem 6. (Hard!) Show that the quotient ring is not a UFD.