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The independence theorem says, roughly, that a family of $\sigma\in\mathrm{Aut}_k(K)$ is $K$-linearly independent. In his Algebra, Serge Lang gave a proof of independence theorem following Artin (which is also the proof used in Jacobson's lecture notes), mentioning that Dedekind used a different approach, with a "suitable choice" of $k$-basis $\omega$ such that $\det(\sigma_i\omega_j)\neq 0$.

I would love to learn more about this choice of $k$-basis. A google search yields Frobenius's determinant theorem. Is there any connection?

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It seems that the original proof can be found in Volume 2 of "Gesammelte matematische Werke" [Complete mathematical works], published in 1930, which was compiled and edited by Fricke, Noether and Ore. More specifically, you can find the full proof -- two pages -- starting on Page 416, Section XLIV. This proof seems to contain the ideas of a suitable choice of $k$-basis with non-zero determinants.

Unfortunately, I am not aware of any existing English translations of this book, but it should be navigable if you are familiar enough in the area, as with all mathematics. The concluding line

... woraus folgt, daß alle $h=0$ sind.

should be recognisable in any proof of linear independence! Note that the full texts of all three volumes of this book are available online from the Göttingen State and University Library, at this link. The page you are interested in is this one.

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    $\begingroup$ Thank you for the reference! It seems to me that Dedekind constructed, by mean of differential forms, a $k$-basis of the subfield $K^H$ consisting of elements fixed under the action of a subgroup $H\in\mathrm{Aut}_k K$ (I guess this subfield is called Ideal), and the group considered is cyclic. It's quite suggestive. $\endgroup$
    – anonymous67
    Commented Feb 21, 2019 at 15:29
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    $\begingroup$ My (admittedly cursory) reading of it agrees with what you say. You may also find it useful to read through this English summarized, annotated and translated version of Dedekind's work on the topic and related material. The proof of the statement can be found on page 39 (where "irreducible" is used in place of "independent"). All needed terminology is helpfully explained in the beginning, so it may be nice to read it all the way from the start. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Feb 21, 2019 at 16:54

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