I don't fully understand what your argument is, but the answer is yes.

In your example, is a field with 9 elements. Let . Then and [K:F] is the Galois extension, where .

Generally speaking, if (p is a prime number here) and , then K is the splitting field of over F (link). Thus K is the Galois extension of F. In fact, , where is a cyclic group generated by defined by for . Verify that has an order n such that .