I wanted to delve into Galois theory as soon as possible, so I started reading a book on the topic without previous knowledge of other areas of mathematics which are often needed to understand Galois. Because of this I do not understand the usual definition of a field extension (which makes use of the concept of a Vector Space), but rather employ a different (I believe equivalent) definition:
Let $L, K$ be subfields of $\mathbb{C}$ such that $K⊆L$
Let $A = $ {$x_1, ..., x_n$} $⊆L$ be a set of linearly independent elements over $K$ such that $L = $ {$x_1k_1+...+x_nk_n|k_i∈K$}, then $[L:K]=n$. We call $A$ a basis for $L:K$.
From such definition, is it possible to prove that if $K⊆M⊆L$, such that $[L:K]=n$ for some $n∈\mathbb{N}$ then $[M:K]$ is well defined. In other words, that there exists a set $B⊆M$ which is a basis for $M:K$.
I'm aware my definition only speaks of finite field extensions.
I would really appreciate any help/thoughts.