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.
  
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'''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.''' (Withdrawn, do not submit) Show that the quotient ring 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.
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Revision as of 10:06, 15 November 2011

In Preparation

The information below is preliminary and cannot be trusted! (v)

This assignment is due at class time on Tuesday, November 29, 2011.

Solve the following questions

Problem 1. Prove that a ring R is a PID iff it is a UFD in which \gcd(a,b)\in\langle a, b\rangle for every non-zero a,b\in R.

Problem 2. (Selick) In a ring R, and element x is called "nilpotent" if for some positive n, x^n=0. Let \eta(R) be the set of all nilpotent elements of R.

  1. Prove that if R is commutative then \eta(R) is an ideal.
  2. Give an example of a non-commutative ring R in which \eta(R) is not an ideal.

Problem 3. (comprehensive exam, 2009) Let A be a commutative ring. Show that a polynomial f\in A[x] is invertible in A[x] iff its constant term is invertible in A and the rest of its coefficients are nilpotent.

Problem 4. (Lang) Show that the ring {\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C} is a PID and hence a UFD. What are the units of that ring?

Problem 5. (Dummit and Foote) In {\mathbb Z}[i], find the greatest common divisor of 85 and 1+13i, and express it as a linear combination of these two elements.

Problem 6. (Hard!) Show that the quotient ring {\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle is not a UFD.