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I'm pretty new to field theory and Galois theory and have been mildly puzzled by the notation $E/F$ to denote $E$ as a field extension of $F$. The notation seems reminiscent to me of quotient ring notation, $R/I$, to denote the ring formed by cosets of an ideal $I$.

However, we can't even form a quotient ring $E/F$ since $F$ is not an ideal in $E$ and the notation $E/F$ doesn't indicate anything similar to a quotient structure.

The "division"-like notation for the quotient ring also made sense, both because of the intuitive idea of "cancelling out" elements from the same ideal as well as the consequences of the third isomorphism theorem which states that $(R/J)/(I/J)\cong R/I$.

The "division"-like notation for field extensions seems to provide no intuition and seems really arbitrary, as well as causes some confusion with objects already denoted this way. Is there a reason this notation is used and is there a deeper concept I may be missing related to all this?

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    $\begingroup$ The notation $E/F$ does not imply an operation being performed on $E$ and $F$. Rather, it's that way simply because "$E/F$" is read as "$E$ over $F$." $\endgroup$ – arctic tern Mar 7 '17 at 4:37
  • $\begingroup$ As J. Rotman says: "the notacion $K/k$ should not be confused with the notation for a quotient ring, for a field $K$ has no interesting ideals; in particular, if $k\subset K$, then $k$ is not an ideal in $K$." $\endgroup$ – Xam Mar 7 '17 at 15:39
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The notation $E/F$ is not confused with quotient rings because quotient rings of fields are trivial since fields don't have nontrivial ideals.

There is no deeper meaning to $E/F$ other than being easy to read as "$E$ over $F$".

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  • $\begingroup$ This does not answer the question "is there a reason this notation is used and is there a deeper concept I may be missing related to all this?" $\endgroup$ – Will R Mar 7 '17 at 12:08

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