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?

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

  • $\begingroup$ The notation $E \leq F$ reads $E$ is a subfield of $F$. $\endgroup$ – dalastboss Jan 26 '16 at 2:46
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    $\begingroup$ In this case, you must have written it backwards. The subfield should be the scalars, not the other way around. Read my paragraph on how the vector space structures works to see why. $\endgroup$ – Future Jan 26 '16 at 2:47

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