I'm taking a field theory course and having trouble keeping the notation straight. In my notes I have written: "if $E, F$ are fields, and $E \leq F$, E is a vector space over F".
First confusion: I take that to mean that the elements of $F$ are (acting as) the scalars and the vectors are $E$?
Later on I have written "$E/F$ is the vector space $E$ with coefficients in $F$", so I think I have this right?
Second confusion: Is this a different construct than an algebraic quotient? For example, for one of my exercises I have to prove a particular property about the structure $F[x]/(f(x))$ ($f$ is a polynomial and $F$ is a field). I take this to mean the cosets of the ideal generated by $f$ in $F[x]$; I don't think this can be equivalently interpreted as a vector space $F[x]$ with coefficients in $(f(x))$. Am I wrong?