2015-33.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	33	IB 2015 Groups, Rings and Modules	Paper 4, Section II, F	2015

Find $a \in \mathbb{Z}{7}$ such that $\mathbb{Z}{7}[x] /\left(x^{3}+a\right)$ is a field $F$. Show that for your choice of $a$, every element of $\mathbb{Z}_{7}$ has a cube root in the field $F$.

Show that if $F$ is a finite field, then the multiplicative group $F^{\times}=F \backslash{0}$ is cyclic.

Show that $F=\mathbb{Z}_{2}[x] /\left(x^{3}+x+1\right)$ is a field. How many elements does $F$ have? Find a generator for $F^{\times}$.