Can one write a finite extension of local fields as a compositum of fields whose degrees are prime powers?

Let $$F/K$$ be a finite extension of local fields of degree $$n$$.

Question: Does there exist intermediate fields $$F/K_i/K$$ such that

• the degree of $$K_i/K$$ is a prime power and
• $$F$$ is the compositum of $$K_1, \dots, K_r$$ for some $$r$$?

My Progress (on an example):

Let $$K=\mathbb{Q}_5$$. Assume $$F/K$$ is an extension such that $$[F:K] = 6 = 2\cdot 3$$. If we also know its ramification degree $$e$$ is $$1$$, then we can find such intermediate fields as follows:

• If $$e=1$$, then the ramification degrees of the intermediate field extensions must be $$1$$ too, i.e. they are unramified. Choose $$K_1 = \mathbb{Q}_5(\xi_{24})$$ and $$K_2 = \mathbb{Q}_5(\xi_{124})$$ where $$\xi_{24}$$ and $$\xi_{124}$$ are a primitive $$24$$-th and a primitive $$124$$-th root of unity, respectively. These extensions are unramified because of the following result:

Proposition 5.4.11 (of Fernando Gouvea's book "$$p$$-adic Numbers - An Introduction)
For each $$f$$ there is exactly one unramified extension of $$\mathbb{Q}_p$$ of degree $$f$$. It can be obtained by adjoining to $$\mathbb{Q}_p$$ a primitive $$(p^f-1)-st$$ root of unity.

• I'm not sure about this one, but it should be $$F=K_1 K_2$$, right?

Thoughts and Ideas which could be useful:

• For the solution, one could maybe use other useful data (e.g. the ramification degree as in the example).
• If we have a local field extension whose degree is a prime, then it is either unramified or unramified. Furthermore, we know already that an unramified degree of arbitrary degree exist. (Is that also true for totally ramified extension?)

Could you please help me with that problem? Any help is appreciated, thank you!

• Totally ramified extension of arbitrary degree also exit, just consider Eisensteni polynomials. – pisco Oct 9 '18 at 10:00