A small observation. By combining this argument with Piotr Achinger's construction, we get an elementary, self-contained proof that there exist transcendental numbers that can be explicitly constructed: namely, the same numbers that Liouville showed to be transcendental in his original proof -- en.wikipedia.org/wiki/…. Maybe this "ultimately" amounts to the same thing as Liouville's proof, but I think I prefer it to the proof on Wikipedia.

Thanks so much, Will -- this clarified things enormously! There was one step that wasn't obvious to me: why does the minimum eigenvalue of the matrix give a lower bound on the actual real valuation of the element? But I think I now see it: let $\alpha_1,\ldots,\alpha_k$ be the generators of the extension field; then $(1,1/\alpha_1,\ldots,1/\alpha_k)$ is always an eigenvector of the matrix corresponding to an element $v\in\mathbb{Q}[\alpha_1,\ldots,\alpha_k]$, and its associated eigenvalue is always $|v|$ (by calculation).

I see; thanks. Is there an easy way to compute the number of such Archimedean valuations?

Timothy: Thanks so much; that's extremely helpful! Just one thing I'm confused about: for a general number field $K$, what are the other Archimedean valuations? So for example, suppose $K=\mathbb{Q}[\sqrt{2},\sqrt{3}]$---then what are the Archimedean valuations besides the usual absolute value?

Thanks so much, Julian! I'm trying to unpack this proof---can anyone suggest a good reference for reading about why the product formula holds for valuations, and why there are only finitely many valuations $v$ for which $|x|_v$ can exceed $1$? Also, for an algebraic number field, how many valuations are there in total---countably many? How do we even see that $\prod_v |x|_v$ is well-defined? (Sorry for asking such basic questions.)