computing the orbits for a group action

abstract-algebra group-theory group-actions

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.


Solution 1:

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

So what about the orbits of the non-fixed points?

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