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Update. It's now on the arXiv.

Some time ago I found my "own" proof of the fundamental theorem of Galois theory. You can find a pdf with the proof (link removed, see arXiv). It is quite short, self-contained, and uses a neat combinatorial argument:

A field cannot be written as a union of finitely many proper subfields

Most users of mathoverflow can simply skip most of it and only read the combinatorial Lemma 3.3. which leads to Prop. 4.2, as well as Lemma 5.5 which leads to to Prop. 6.3. The rest is easy.

I wonder if this proof is new or not. For sure I have never seen it before, and I checked a bit the literature and couldn't find it so far. But also I am not really an expert on the history of algebra at all, and my days as an "active mathematician" are over anyway. Hopefully somebody else has a better overview?

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    $\begingroup$ +1. Let me mention the link math.stackexchange.com/a/89576/660 even if it doesn't answer your question. $\endgroup$
    – Pierre-Yves Gaillard
    Commented Nov 4, 2022 at 19:09
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    $\begingroup$ I 've definitely seem that a vector space cannot be written as a finite union of proper subspaces to prove prove the primitive element theorem $\endgroup$
    – Benjamin Steinberg
    Commented Nov 4, 2022 at 21:23
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    $\begingroup$ The "neat combinatorial argument" is in Bourbaki, algebre, ch. V.40 (the section on the primitive element). $\endgroup$
    – coudy
    Commented Nov 8, 2022 at 19:24
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    $\begingroup$ About Lemma 3.2: it's contained in a result of B.H. Neumann (Groups covered by finitely many cosets. Publ. Math. Debrecen 3 (1954), 227-242) which says that if a group is covered by finitely many cosets, it's covered by just those of finite index. In particular if the cover is not redundant, then all have finite index. So, if a group is covered in a non-redundant way by finitely many subgroups $G_i$, then the intersection $\bigcap G_i$ has finite index. $\endgroup$
    – YCor
    Commented Nov 15, 2022 at 7:28
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    $\begingroup$ That your days as an active mathematician are over probably means that you continue your career outside academia. Good luck, Martin! I hope you are staying active in mathematics and, in particular, on MO. $\endgroup$
    – Jochen Wengenroth
    Commented Nov 15, 2022 at 9:07

1 Answer 1

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At a first glance your approach reminds me of Meinolf Geck's American Mathematical Monthly article, see also the arxiv version of his article.

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  • $\begingroup$ This is a very interesting paper, thank you! $\endgroup$
    – Martin Brandenburg
    Commented Nov 8, 2022 at 20:36
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    $\begingroup$ That is the paper I was going to mention too. $\endgroup$
    – KConrad
    Commented Nov 8, 2022 at 21:30
  • $\begingroup$ I have now mentioned this paper in the current version of the pdf. $\endgroup$
    – Martin Brandenburg
    Commented Nov 11, 2022 at 21:25
  • $\begingroup$ Although this is not a direct answer to my question, it is as close as it can get, I guess. The reaction here (and my further research) seems to indicate that there are proofs related to mine, but not the same. So I will accept it. Thank you! $\endgroup$
    – Martin Brandenburg
    Commented Nov 15, 2022 at 2:03

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