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.

  • $\begingroup$ Do you know the number of fixed points and the order of your group? $\endgroup$ – j.p. Mar 15 '15 at 10:09
  • $\begingroup$ 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. $\endgroup$ – 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 $\mathbb F_3$).

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

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