*Update.* It's now on the [arXiv](https://arxiv.org/abs/2211.07508).




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Some time ago I found my "own" proof of the [fundamental theorem of Galois theory](https://en.wikipedia.org/wiki/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?



A simple proof of the fundamental theorem of Galois theory
At a first glance your approach reminds me of [Meinolf Geck's](https://www.jstor.org/stable/10.4169/amer.math.monthly.121.07.637#metadata_info_tab_contents) American Mathematical Monthly article, see also the [arxiv version](https://arxiv.org/abs/1306.3853) of his article.


