Question 9(b) Section- Fields and Galois Theory Sub section- Fundamental theorem of Galois Theory

Question

I am trying some problems from Hungerford Algebra in Galois Theory.

I have a question in problem 9 whose image I am adding( I have also added statement of problem 6 in case it's useful) :

I have proved (a).

But problem arises in (b) .

I took $|K|= n => Aut_K{ K(u) }\leq n^4$.

And $ |{1_K}':{Aut_{K}{K(x) }}'| \leq | Aut_{K} K(x) : 1_K| $

Now,as K(x) is Galois over K it implies $|K(x):K|\leq | Aut_{K} K(x)|\leq n^4$ .

But I am not able to see any contradiction.

Can you please tell how to get a contradiction in (b).