Abstract Algebra: Theory and Applications

What is the order of the element $3$ in $U(20)\text{?}$

What is the order of the element $5$ in $U(23)\text{?}$

Find three generators of ${\mathbb{Z}}_8\text{.}$

Find three generators of the $5^{\text{th}}$ roots of unity.

Show how to compute ${15}^{40}(\mathrm{mod}23)$ efficiently by hand. Check your answer with Sage.