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Greetings to everyone on this forum (I am a new-comer). I would like to ask the experienced members for suggestions on (as) comprehensive and systematic (as possible) bibliographic sources regarding:

  • the theory of ordered fields respectively valuation fields
  • Puiseux and Levi-Civita formal series and their properties
  • more general Hahn-Mal'cev constructions with their algebraic & order properties.

I am aware of a number of disparate articles and textbooks dealing with some aspects of the theories listed above, however I do not know of any systematic, compendium or monograph-style reference on these topics.

I avidly appreciate the detail and meticulousness in monographs, so please do not shy away from mentioning such items if they might happen to exist.

With regards for those having taken their time to read this〜

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    $\begingroup$ My two cents: the only (very few) things I know on Levi-Civita formal series (and more generally on Levi-Civita fields) I learned from reading the paper "Tullio Levi-Civita's Work on Nonarchimedean Structures (with an Appendix: Properties of Levi-Civita Fields)", in Tullio Levi-Civita. Convegno Internazionale Celebrativo della Nascita (Rome, 17-19 dicembre 1973), Atti dei Convegni Lincei, 8, Rome: Accademia Nazionale dei Lincei, 1975, pp. 297-312. It is a very clear and elementary survey paper on Levi-Civita's contributions to the field of formal power series. I hope this helps $\endgroup$
    – Daniele Tampieri
    Dec 16, 2020 at 7:08
  • $\begingroup$ @DanieleTampieri Thank you kindly for your suggestion! $\endgroup$
    – ΑΘΩ
    Dec 16, 2020 at 8:17
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    $\begingroup$ ΑΘΩ, you are welcome. By the way, I forgot to mention in my previous comment that the Author of the cited paper is Detlef Laughwitz. $\endgroup$
    – Daniele Tampieri
    Dec 16, 2020 at 8:34
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    $\begingroup$ For valued fields whose value group is a subgroup of $\mathbb{R}$, a very good reference is Paolo Ribenboim's The Theory of Classical Valuations. For general valued fields and also for ordered valued fields, a good reference is the third chapter of Aschenbrenner, van den Dries and van der Hoeven's Asymptotical Differential Algebra and Model Theory of Transseries. This last one also contains some notes about Hahn series fields, although a more complete reference for their role in general valuation theory is Irving Kaplansky's Maximal fields with valuations I and II. $\endgroup$
    – nombre
    Dec 16, 2020 at 9:12
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    $\begingroup$ @DanieleTampieri You are most kind to have taken the time to find a solution to this matter! I will try to see if the Atti can be found somewhere locally near me, and if not I will get in touch with you at the email address you have specified. With regards〜 $\endgroup$
    – ΑΘΩ
    Jan 30, 2021 at 12:12

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