In a previous question, Algebra Fan said:
In general, if you have two fields $K\subset L$, $L$ can be viewed as a $K$− vector space with the obvious scalar multiplication: if $k\in K$ and $\ell \in L$, then $k⋅\ell =k\ell$ where $k\ell$ is just the product of $k$ and $\ell$.
I've been thinking about this for a while, and I'm not sure I understand it.
We say $V$ is a vectorspace over $R$ if it follows some rules, including that there be some multiplication operation $R\times V\to V$. Some questions:
- Is the analogous multiplication operation in a general field $K\times L \to L$ or $K\times L\to K$?
- Is the dimension of the vectorspace the order of $K$? The order of $L$? The index of $K$ in $L$?
- Are the basis units just the elements of $K$?
Alternatively, if someone could link me to lecture notes etc. giving some intuition about what's going on that would be appreciated.