Question

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

Answer 1

Answer:

Since 3 elements are fixed so there are 3 singleton orbits. Number of elements in other orbits are divisors (other than 1) of |Gal(F9/F3)|, i.e. 2. So there are 3 orbits containing 2 elements. Hence total number of orbits =3+3=6.