| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
34 |
IB |
2012 |
Groups, Rings and Modules |
|
Paper 3, Section II, G |
2012 |
For each of the following assertions, provide either a proof or a counterexample as appropriate:
(i) The ring $\mathbb{Z}_{2}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.
(ii) The ring $\mathbb{Z}_{3}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.
(iii) If $F$ is a finite field, the ring $F[X]$ contains irreducible polynomials of arbitrarily large degree.
(iv) If $R$ is the ring $C[0,1]$ of continuous real-valued functions on the interval $[0,1]$, and the non-zero elements $f, g \in R$ satisfy $f \mid g$ and $g \mid f$, then there is some unit $u \in R$ with $f=u \cdot g$.