14-1100/Homework Assignment 4

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This assignment is due at class time on Thursday, November 20, 2011.

Solve the following questions

Problem 1. (Klein's 1983 course)

  1. Show that the ideal I=\langle 3,\, x^3-x^2+2x-1\rangle inside the ring {\mathbb Z}[x] is not principal.
  2. Is {\mathbb Z}[x]/I a domain?

Problem 2. 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 3. (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 4. (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 5. (Klein's 1983 course) Show that {\mathbb Z}[\sqrt{10}] is not a UFD.

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