# Notation: quotient ring versus vector space

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?

• Do you mean $F$ is a vector space over $E$? – Future Jan 26 '16 at 2:31
• I'm not sure which part you're correcting? Is that how I am to read $E/F$? Or is that a consequence of $E \leq F$? – dalastboss Jan 26 '16 at 2:32

Definition: If $E$ is a field containing the subfield $F$, then $E$ is said to be an extension field of $F$ denoted as $E/F$. This is not the quotient of $E$ by $F$.
In this case, as $F$ injects into $E$, the natural multiplication structure of $E$ makes $E$ into a vector space over $F$, where the action is just considering elements of $F$ as elements of $E$ and then performing the multiplication in $E$.
I am not sure what $E \subseteq F$ notation means. Although I personally never use it, I have seen people use it in ring theory to mean "subring" or "subfield." Either interpretation would be wrong in this case, because we are looking at $F$ as a subfield of $E$.
• The notation $E \leq F$ reads $E$ is a subfield of $F$. – dalastboss Jan 26 '16 at 2:46