# 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?)