2004-76.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	76	IB 2004 Groups, Rings and Modules	4.II.12F	2004

Answer the following questions, fully justifying your answer in each case.

(i) Give an example of a ring in which some non-zero prime ideal is not maximal.

(ii) Prove that $\mathbb{Z}[x]$ is not a principal ideal domain.

(iii) Does there exist a field $K$ such that the polynomial $f(x)=1+x+x^{3}+x^{4}$ is irreducible in $K[x]$ ?

(iv) Is the ring $\mathbb{Q}[x] /\left(x^{3}-1\right)$ an integral domain?

(v) Determine all ring homomorphisms $\phi: \mathbb{Q}[x] /\left(x^{3}-1\right) \rightarrow \mathbb{C}$.