Consider the following statement:

Assume $E$ and $F$ are unramified (over some fixed prime) finite separable extensions of a field $K$. Then $EF$ is also unramified.

I always thought that it is called Abhyankar's lemma (and indeed it is a special case of it, see https://en.wikipedia.org/wiki/Abhyankar%27s_lemma).

My question is whether this statement is indeed named after Abhyankar. If so, a reference would be appreciated, if not, how is it called?

  • 1
    $\begingroup$ Please clarify what you mean by "prime". Are you fixing a dvr $V$ with fraction field $K$ and then picking compatible maximal ideals over that of $V$ in the integral closure of $V$ in each of $E$, $F$, and (a choice of compositum) $EF$ (the notation $EF$ not being uniquely determined otherwise without a Galois condition), or working with all maximal ideals of the integral closure of $V$ in $EF$ (and the contractions of such to $E$ and $F$)? Whatever it is, this shouldn't be named after Abhyankar, since his lemma is all about how to go beyond this much more classical case. $\endgroup$ – nfdc23 Jan 30 '18 at 18:14
  • $\begingroup$ @nfdc23 Sorry for not being so precise, both are OK (I meant all the maximal ideals). Also, I was under the implicit assumption that $E$ and $F$ are contained in a joint over field. In any rate, your last remark is not clear to me. Even if it is easy nowadays, still it was done by someone. And even if Abhyankar proved a more general statement, he might have been the first to prove this simpler (but important) statement. $\endgroup$ – Lior Bary-Soroker Jan 30 '18 at 20:13
  • $\begingroup$ There is no possibility Abhyankar was the first to prove the preservation of unramifiedness for composites. This had to be known since the early days of local fields since it is bound up with existence of "maximal unramified extension"; e.g., after reducing to the complete case (tensoring to completion of $K$ throughout) we invoke the existence of a unique "maximal unramified subextension" (modern references are end of III.5 in Serre's Local Fields, or Thm. 2 + Cor. 2 in section 7 of Ch. I in Cassels-Frohlich, or ...) to conclude the one for $EF/K$ contains $E/K$ and $F/K$, hence is $EF$. $\endgroup$ – nfdc23 Jan 30 '18 at 22:08
  • $\begingroup$ @nfdc23 my question is to who to attribute this. The reference I had in mind was to Abhyankar, and I want to know if there is an earlier one. $\endgroup$ – Lior Bary-Soroker Jan 30 '18 at 22:25
  • $\begingroup$ Attribution may hard, but it must vastly predate Abhyankar. A lot of facts in the ramification theory of local fields were known in some way to the early 20th century number theorists, and it could be hard to definitively attribute such a fact to a specific person. For Hensel's Lemma we have a name, but for preservation of unramifiedness under the formation of composite fields (one of the first issues that comes to mind after recognizing the usefulness of unramified extensions) it might be hard to know. Hilbert defined higher ramification groups, so maybe look in his Zahlbericht? $\endgroup$ – nfdc23 Jan 31 '18 at 0:36

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