Talk:Levi-Civita field

Headers

Someone else will have to add the appropriate headers; please delete this line when done. — Arthur Rubin (talk) 11:44, 19 January 2016 (UTC)

Huh? 67.198.37.16 (talk) 00:44, 6 July 2016 (UTC)

Similar fields

The present definition

$\sum _{q\in \mathbb {Q} }a_{q}\varepsilon ^{q},$

where $a_{q}\,$ are real numbers, $\mathbb {Q}$ is the set of rational numbers, and $\varepsilon$ is to be interpreted as a positive infinitesimal. The restriction that the support is left-finite, * $\{q\in \mathbb {Q} :a_{q}\neq 0\&q<r\}$ is finite for any r, can be replaced by other similar conditions:

1. $\{q\in \mathbb {Q} :a_{q}\neq 0\}$ is bounded below, and there is a positive integer n such that $\forall q\in \mathbb {Q} :nq\in \mathbb {Z}$
   • That is, $\mathbb {F} =\bigcup _{n\in \mathbb {N} }\mathbb {R} ((\varepsilon ^{\frac {1}{n}}))$
   • This is also known as the field of Puiseux series
2. $\{q\in \mathbb {Q} :a_{q}\neq 0\}$ is well-ordered
   • This relates to the Hahn series
3. The coefficient field $\mathbb {Q}$ can be replaced by any totally ordered divisible group.
   • The simplest example would be replacing it by $\mathbb {R}$, but replacing it by $\mathbb {Q} \times \mathbb {Q}$ may also have some interest.

If I can find literature discussing these fields, should it be added to the article? — Arthur Rubin (talk) 11:44, 19 January 2016 (UTC)

Well, given that we already have long articles on the puiseaux and hahn series, then certainly 1 and 2 are very appropriate for this article. I'm don't know why you are even asking, these seem obviously appropriate for article inclusion. Point three also seems appropriate, but should get another sentence sketching what it is about total orders that is needed to have a workable definition. 67.198.37.16 (talk) 16:26, 5 July 2016 (UTC)

Properties

This publication, already referenced in this article, makes a number of interesting, important assertions and clarifications: viz. the Levi-Civita field is the smallest extension of thr reals that is both... umm, let me rephrase: The ring of polynomials $\mathbb {R} [[X]]$ is not a field. The smallest field that contains it is the field of fractions (Laurent series) $\mathbb {R} ((X))$ However, the field of fractions is not real closed nor Cauchy complete. The real closure is the Puiseux series, but the Puiseux series is not Cauchy complete. The smallest field that is Cacuhy complete (in the order topology) is .. you guessed it ... the Levi-Civita field. I think that is just an awesome fact/result, and makes it just ideal for non-standard analysis, among other things, which is something that essentially everything I've ever skimmed on non-standard analysis seems to have completely missed. Soo ... adding the above to this article would be excellent. 67.198.37.16 (talk) 00:30, 6 July 2016 (UTC)

Is the Levi-Civita field over an algebraically closed field algebraically closed?

If K is an algebraically closed field (not necessarily of characteristic zero), is the Levi-Civita field with coefficients in K algebraically closed? 129.104.241.193 (talk) 23:41, 30 May 2024 (UTC)

Ah, if K is characteristic zero then the answer is yes, because the Levi-Civita field is the completion of the field of Puiseux series, and the latter is algebraically closed.

If K has positive characteristic zero then the answer is no, as shown by the example given in the article Hahn series (the set of exponents

$\left\{-{\frac {1}{p}},-{\frac {1}{p^{2}}},-{\frac {1}{p^{3}}},\ldots \right\}$

has infinitely many elements below 0). 129.104.241.193 (talk) 23:52, 30 May 2024 (UTC)