I propose the removal of the tag extension-field. There are currently a grand total of 39 questions tagged with it so it would not take long to remove it. Furthermore I don't think the tag is particularly useful, field-theory or galois-theory almost always suffices. See for yourself here.https://math.stackexchange.com/questions/tagged/extension-field

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    $\begingroup$ Often the questions covered by tag a are a subset of the questions covered by tag b, but tag a is still useful to filter the questions further when searching for a particular topic. $\endgroup$ – robjohn Mar 7 '13 at 17:16
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    $\begingroup$ This could be a post in the tagging thread meta.math.stackexchange.com/questions/1363/… (not necessary to open a thread for each tag... there are so many of those). $\endgroup$ – user53153 Mar 7 '13 at 17:25
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    $\begingroup$ Seems to me that we could profitably just make it an synonym for field-theory. Is there any nontrivial part of field theory that is not about field extensions anyway? $\endgroup$ – hmakholm left over Monica Mar 7 '13 at 18:33
  • $\begingroup$ I share your feelings about this tag. If judged prudent/helpful, I could volunteer to do a retagging-spree getting rid of it. $\endgroup$ – Jyrki Lahtonen Mar 18 '13 at 10:36

I don't think the tag should be removed: there is an important difference between a field extension of a field $k$ and an arbitrary algebra over $k$.
For example a commutative $k$-algebra $A$ is defined to be separable if for any field extension $E$ of $k$ the $k$-algebra $E\otimes_k A$ is reduced.
Extensions $E$ of $k$ play a vital role here in the study of $k$-algebras which need not be extensions of $k$: for example it is enough to test reducedness for $E=k^{p^{-\infty}}$, a perfect closure of $k$.
Also transcendental field extensions are quite important in algebraic geometry and they have nothing to do with Galois theory, so that the tag galois-theory would not be a suitable replacement.
And let me also mention, as an answer to Henning's question, that there are non-trivial results of field theory which are not about extension fields: for example that the tensor products of two reduced algebras over a perfect field is reduced, a fact that was asked about on our site.

  • $\begingroup$ I understand your point. But that's certainly not how the tag is used on this site. It tends to be used in such questions as show that $[\mathbb Q(\sqrt{3},\sqrt{2}):\mathbb Q]=4$. $\endgroup$ – JSchlather Mar 19 '13 at 5:30

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