(a) Let $K$ be a field. State what it means for $\xi_{n} \in K$ to be a primitive $n$th root of unity.

Show that if $\xi_{n}$ is a primitive $n$th root of unity, then the characteristic of $K$ does not divide $n$. Prove any theorems you use.

(b) Determine the minimum polynomial of a primitive 10 th root of unity $\xi_{10}$ over $\mathbb{Q}$.

Show that $\sqrt{5} \in \mathbb{Q}\left(\xi_{10}\right)$.

(c) Determine $\mathbb{F}_{3}\left(\xi_{10}\right), \mathbb{F}_{11}\left(\xi_{10}\right), \mathbb{F}_{19}\left(\xi_{10}\right)$.

[Hint: Write a necessary and sufficient condition on $q$ for a finite field $\mathbb{F}_{q}$ to contain a primitive 10 th root of unity.]