The Joly–Becker theorem for $*$–orderings

Igor Klep; Dejan Velušček

Displaying similar documents to “The Joly–Becker theorem for $*$ –orderings”

Subfields of henselian valued fields

Ramneek Khassa, Sudesh K. Khanduja (2010)

Colloquium Mathematicae

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Let (K,v) be a henselian valued field of arbitrary rank which is not separably closed. Let k be a subfield of K of finite codimension and $v_{k}$ be the valuation obtained by restricting v to k. We give some necessary and sufficient conditions for $(k, v_{k})$ to be henselian. In particular, it is shown that if k is dense in its henselization, then $(k, v_{k})$ is henselian. We deduce some well known results proved in this direction through other considerations.

On the subfields of cyclotomic function fields

Zhengjun Zhao, Xia Wu (2013)

Czechoslovak Mathematical Journal

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Let $K = 𝔽_{q} (T)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f (T) \in 𝔽_{q} [T]$ with positive degree, denote by $Λ_{f}$ the torsion points of the Carlitz module for the polynomial ring $𝔽_{q} [T]$ . In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K (Λ_{P})$ of degree $k$ over $𝔽_{q} (T)$ , where $P \in 𝔽_{q} [T]$ is an irreducible polynomial of positive degree and $k > 1$ is a positive divisor of $q - 1$ . A formula for the analytic class...

On $p^{2}$ -Ranks in the Class Field Tower Problem

Christian Maire, Cam McLeman (2014)

Annales mathématiques Blaise Pascal

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Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^{2}$ -rank of the class group as a quantity of relevance in the $p$ -class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class...

Relatively complete ordered fields without integer parts

Mojtaba Moniri, Jafar S. Eivazloo (2003)

Fundamenta Mathematicae

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We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series $[[F^{G}]]$ with exponents in a totally ordered Abelian group G and coefficients in an ordered field F. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that $[[F^{G}]]$ is always Scott complete. In contrast, the Puiseux...

O-minimal version of Whitney's extension theorem

Krzysztof Kurdyka, Wiesław Pawłucki (2014)

Studia Mathematica

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This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic $^{p}$ -Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a $^{p}$ -function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a...

Some notes on the composite $G$ -valuations

Angeliki Kontolatou (1994)

Archivum Mathematicum

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In analogy with the notion of the composite semi-valuations, we define the composite $G$ -valuation $v$ from two other $G$ -valuations $w$ and $u$ . We consider a lexicographically exact sequence $(a, β) : A_{u} \to B_{v} \to C_{w}$ and the composite $G$ -valuation $v$ of a field $K$ with value group $B_{v}$ . If the assigned to $v$ set $R_{v} = {x \in K / v (x) \geq 0$ or $v (x)$ non comparable to $0}$ is a local ring, then a $G$ -valuation $w$ of $K$ into $C_{w}$ is defined with its assigned set $R_{w}$ a local ring, as well as another $G$ -valuation $u$ of a residue field is defined with $G$ -value group $A_{u}$ . ...

Greatest common divisors of $u - 1$ , $v - 1$ in positive characteristic and rational points on curves over finite fields

Pietro Corvaja, Umberto Zannier (2013)

Journal of the European Mathematical Society

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In our previous work we proved a bound for the $g c d (u - 1, v - 1)$ , for $S$ -units $u, v$ of a function field in characteristic zero. This generalized an analogous bound holding over number fields, proved in [3]. As pointed out by Silverman, the exact analogue does not work for function fields in positive characteristic. In the present work, we investigate possible extensions in that direction; it turns out that under suitable assumptions some of the results still hold. For instance we prove Theorems 2 and 3...

On the Stark-Shintani units and the ideal class groups in the cyclotomic $ℤ_{p}$ -extensions of class fields over real quadratic fields

Tsuyoshi Itoh (2002)

Acta Arithmetica

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Governing fields for the 2-classgroup of $Q (s q r t - q_{1} q_{2} p)$ and a related reciprocity law

Patrick Morton (1990)

Acta Arithmetica

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Displaying similar documents to “The Joly–Becker theorem for * –orderings”

Displaying similar documents to “The Joly–Becker theorem for $*$ –orderings”