# computing the orbits for a group action

Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Then the number of orbits for the action of $G$ on the fields with nine elements is

1. 3

2. 5

3. 6

4. 9

I have no idea how to compute the numbers of orbits for a group action. Anyone please help me. Thanks.

• Do you know the number of fixed points and the order of your group?
– j.p.
Mar 15 '15 at 10:09
• If you like to use a bit more of Galois theory instead, you could look also at the degrees of the minimal polynomials of all elements of $\mathbb{F}_9$ over $\mathbb{F}_3$. The Galois group acts transitively on roots of the minimal polynomials.
– j.p.
Mar 15 '15 at 11:48

Hint The Frobenius is an involution on $\mathbb F_9$ with $3$ fixed points (the elements of $\mathbb F_3$).