Let G be the Galois group of a field with nine elements over its subfield with three elements. Then find the number of orbits for the action of G on the field with 9 elements.
Clearly $|Gal(\mathbb F_9/\mathbb F_3)|=2$. Since $3$ elements are fixed so there are $3$ singleton orbits. Number of elements in other orbits are divisors (other than $1$) of $|Gal(\mathbb F_9/\mathbb F_3)|$, i.e. $2$. So there are $3$ orbits containing $2$ elements.
Hence total number of orbits $=3+3=6$.