If $I_m$ denotes the set of integers modulo $m$, then the following are fields with respect to the operations of addition modulo $m$ and multiplication modulo $m$:
$(i) Z_{23}$
$(ii) Z_{29}$
$(iii) Z_{31}$
$(iv) Z_{33}$
My attempt:
Answers is given $(i), (ii),\space\text{and}\space(iii)$.
I have read upto group from Rosen. Sorry, I did not read field that was out of syllabus. This question was from UGC-NET-2004-ii-CS paper.
Can you explain it, please?
Basically, a field is a thing where you can add, subtract, multiply and divide. It is a bit tricky to see that the first three examples ($\mathbb{Z}_{23}$, $\mathbb{Z}_{29}$, $\mathbb{Z}_{31}$) are indeed fields. In fact, $\mathbb{Z}_{p}$ happens to be a field always when $p$ is prime, and this result follows from Fermat's little theorem.
But let us look at the fourth example. Assume you can divide the elements by $11$, then you have $$3=\frac{11}{11}\cdot3=\frac{11\cdot3}{11}=\frac{33}{11}=0,$$ a contradiction. (The latter equality holds because $33=0$ modulo $33$.) A similar argument shows you that $\mathbb{Z}_q$ cannot be a field if $q$ is any composite number.