Calculating the order of the multiplicative group $k^\times$

Question

Let $k$ be a field of cardinality $64$.

1. What is the order of the multiplicative group $k^\times$ of invertible elements of $k$? What are the possible orders of elements of $k^\times$?

2. Let $d\geq0$ be an integer. Show that $k^\times$ contains at most $d$ elements of order $d$.

3. Hence prove that $k^\times$ is a cyclic group.

What I have tried so far:

I have determined that the order of the multiplicative group is 63. Hence the possible orders are the factors of 63.

I know that for question 3, a group of order n is cyclic iff it contains an element of order n.

Answer 1

Hints.

1. $k$ is a field. How many elements does $k^\times$ have?

2. What equation does an element of order $d$ satisfy?

3. Add up the number of elements by order.