Finding subfields of a field extension and help with the definition

Question

Let K ⊂ L an extension with [L : K] = p a prime number. What are the subfields of L containing K?

Definition says that: Let K be a field. An extension of K is a field L such that K ⊂ L. Then L is naturally a K-vector space. The extension K ⊂ L is said to be finite whenever the dimension dim_k(L) is finite. In this case, we denote by [L:K] this dimension: it is the degree of the extension.

1. What will it say that: L is naturally a K-vector space.
2. What will it say that: the dimension dim_k(L) is finite

I am thinking that the subfields of L containing K is : Q, R, C and Z_p, since they you got prime numbers . Think i need subfields with prime numbers because the degree is a prime number, but i dont see the pattern to solve this type of problems.

Answer 1

let $K\subseteq F\subseteq L$ be an intermediate field. We know that: $$\left[L:F\right]\cdot\left[F:K\right]=\left[L:K\right]$$ and so: $$\left[L:F\right]|\left[L:K\right]=p$$ since p is prime we get: $$\left[L:F\right]\in\left\{ 1,p\right\} $$ and so: $F=L\,or\,F=K$