# 14-1100/Homework Assignment 4

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

Problem 2. Prove that a ring ${\displaystyle R}$ is a PID iff it is a UFD in which ${\displaystyle \gcd(a,b)\in \langle a,b\rangle }$ for every non-zero ${\displaystyle a,b\in R}$.

Problem 3. (Lang) Show that the ring ${\displaystyle {\mathbb {Z} }[i]=\{a+ib\colon a,b\in {\mathbb {Z} }\}\subset {\mathbb {C} }}$ is a PID and hence a UFD. What are the units of that ring?

Problem 4. (Dummit and Foote) In ${\displaystyle {\mathbb {Z} }[i]}$, find the greatest common divisor of ${\displaystyle 85}$ and ${\displaystyle 1+13i}$, and express it as a linear combination of these two elements.

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

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