In a number field that is not necessarily totally real, could it make sense to consider "totally positive" elements as elements that are positive in all real embeddings? (So in a totally complex field for example, every element would be vacuously totally positive). Does anyone actually use this? Would it be misleading to call such elements totally positive?

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    $\begingroup$ Yes, absolutely, I would say this is common use. And according to the first sentence in Siegel's 1921 paper "Darstellung total positiver Zahlen durch Quadrate", apparently already Hilbert used it in that sense. $\endgroup$ – Arno Fehm Oct 16 at 16:29
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    $\begingroup$ Maybe one should add that totally positive elements in this sense are always precisely the sums of squares, by a theorem of Artin and Schreier. $\endgroup$ – Arno Fehm Oct 16 at 16:34
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    $\begingroup$ Section 11.3 of Jacobson's Basic Algebra II (2nd ed.) is titled "Totally Positive Elements" and it proves part of the result of Landau and Siegel (conjectured by Hilbert) on sums of squares in number fields that Arno Fehm mentions in a comment above, but Jacobson doesn't include the more refined information about the number of squares needed. For that, see mathoverflow.net/questions/14456/…. $\endgroup$ – KConrad Oct 16 at 20:34

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