Timeline for Field complete with respect to inequivalent absolute values
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| Nov 5, 2022 at 15:51 | comment | added | KConrad | Yes. I had in mind the original question posted by Yuri, about complete fields, when I said being separably closed implies being algebraically closed. So I should not have written that Schmidt's theorem as you wrote it can have the separable closedness condition made stronger, since Schmidt's theorem is about more general fields than complete ones (namely Henselian ones). | |
| Nov 5, 2022 at 14:21 | comment | added | Arno Fehm | That you so much for the enlightening remark, @KConrad! Just to clarify: While it indeed seems that for complete absolute values the only possibility are the algebraically closed fields, the theorem of F.K.Schmidt as stated in my answer - for henselian rather than complete - can not be strengthened from separably closed to algebraically closed. I guess that is what you meant (and is explained also on the mentioned page 40 in Roquette's paper). | |
| Nov 5, 2022 at 13:48 | comment | added | KConrad | Moreover, from the top of page 40 in Roquette's paper, separably closed implies algebraically closed for complete valued fields: in char. $p$, $x^p - a$ for nonzero $a$ can be approximated by the separable $x^p-cx-a$ for small nonzero $c$, and let $c \to 0$ to get a root of $x^p-a$ as a limit. So Schmidt's theorem that you cite can be strengthened in its conclusion to say $K$ is algebraically closed, which is exactly the kind of counterexample offered in other answers here. So "This is, however, essentially the only possibility" in your answer can be refined to "This is the only possibility". | |
| Nov 5, 2022 at 13:45 | comment | added | KConrad | Section 4.2 of part 1 of Roquette's nice historical survey on valuation theory (see mathi.uni-heidelberg.de/~roquette/hist_val.pdf) shows that F. K. Schmidt had posed the uniqueness question Yuri asked here, for non-algebraically closed fields. On page 39 is an excerpt from a letter Schmidt sent to Hasse in 1930: "Next I wish to investigate whether a field could be complete with respect to two different valuations, which of course cannot be discrete. And if this is possible then I want to characterize the structure of such fields." | |
| Nov 5, 2022 at 6:58 | history | edited | Arno Fehm | CC BY-SA 4.0 |
added 117 characters in body
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| Nov 5, 2022 at 6:51 | history | answered | Arno Fehm | CC BY-SA 4.0 |