Groups, Rings and Modules | Part IB, 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 Z[x]\mathbb{Z}[x] is not a principal ideal domain.

(iii) Does there exist a field KK such that the polynomial f(x)=1+x+x3+x4f(x)=1+x+x^{3}+x^{4} is irreducible in K[x]K[x] ?

(iv) Is the ring Q[x]/(x31)\mathbb{Q}[x] /\left(x^{3}-1\right) an integral domain?

(v) Determine all ring homomorphisms ϕ:Q[x]/(x31)C\phi: \mathbb{Q}[x] /\left(x^{3}-1\right) \rightarrow \mathbb{C}.

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