Field Of Power Series Research Articles

On wild division algebras over fields of power series

Certain special classes of division algebras over the field of Laurent power series with arbitrary residue field are studied. We call algebras in these classes split and well-split algebras. These classes are shown to contain the group of tame division algebras. For the class of well-split division algebras we prove a decomposition theorem which is a generalization of the well-known decomposition theorems of Jacob and Wadsworth for tame division algebras. For both classes we introduce the notion of a -map and develop the technique of -maps for division algebras in these classes. Using this technique we prove decomposition theorems, reprove several old well-known results of Saltman, and prove Artin's conjecture on the period and index in the local case: the exponent of a division algebra over a -field is equal to the index of if , where is a -field. In addition we obtain several results on split division algebras, which, we hope, will help in further research of wild division algebras.

Value groups, residue fields, and bad places of rational function fields

We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field K(x) in one variable, we consider the relative algebraic closure of K in the henselization of K(x) with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of K. In the tame case, we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the p-adics.

Zur Einbettung uniform bewerteter Tern�rk�rper in Hahn-Tern�rk�rper

In this paper, we generalize the results of [12] and derive criteria for the regular embeddability of a uniformly valued ternary field into an appropriate Hahn ternary field of formal power series with coefficients in the residue ternary field and exponents in the value loop. Furthermore, we discuss these criteria also for richer algebraic structures and we give an example for the skew field case.

Valuations in fields of power series

This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.

Artin–Schreier extensions and Galois module structure

Let k be the power series field over a finite field of characteristic p>0. Let L be a cyclic extension over k of degree p. We give a necessary and sufficient condition for the integer ring O L to be free over the associated order. Moreover, when O L is free, we construct a free basis explicitly.

Projective Schur groups of iterated power series fields

The Brauer–Witt Theorem states that every Schur algebra over a field K is Brauer equivalent to a cyclotomic algebra. A central conjecture on the projective Schur group of a field is the analogue of this theorem, which asserts that every projective Schur algebra over a field K is Brauer equivalent to a radical algebra. The conjecture is so far known to be true in characteristic p and for local and global fields. The next natural class of fields to test is power series fields over local and global fields. In this paper we verify the conjecture for these fields and more generally for iterated power series fields over local and global fields.

Symmetry of Sections in Fields of Formal Power Series and a Non-Standard Real Line

Let \(R\left[ {\left[ {G,\beta } \right]} \right]\) be a field of formal power series with real coefficients, whose supports are well ordered subsets of an Abelian group \(G\) of cardinality strictly less than \(\beta \). For \(R\left[ {\left[ {G,\beta } \right]} \right]\), we give criteria of a section being symmetric and of a symmetric section being Dedekind. It is proved that an \(\alpha ^ + \)-saturated non-standard real line \({}^*R\) is isomorphic to some field of the form \(R\left[ {\left[ {G,\alpha ^ + } \right]} \right]\). For \({}^*R\), some consequences are inferred regarding symmetric sections, and the cofinality of “banks” of the sections.

COMPUTING LEADING EXPONENTS OF NOETHERIAN POWER SERIES

ABSTRACT For a field of characteristic zero, we study the field of Noetherian power series, , which consists of maps whose supports are Noetherian (i.e., reverse well-ordered) subsets of . There is a canonical valuation that sends a nonzero series to the maximum element of its support. Given a nonzero polynomial and a series that is transcendental over , we construct a formula for in terms of the roots of . Using this formula, we find sufficient conditions for to be a well-ordered subset of . In particular, this set is well-ordered in case the support of consists solely of positive numbers.

Uncertainty relations in deformation quantization

Robertson and Hadamard-Robertson theorems on non-negative definite hermitian forms are generalized to an arbitrary ordered field. These results are then applied to the case of formal power series fields, and the Heisenberg-Robertson, Robertson-Schrodinger and trace uncertainty relations in deformation quantization are found. Some conditions under which the uncertainty relations are minimized are also given.

Analytic functions over a field of power series

We extend the notion of absolute convergence for real series in several variables to a notion of convergence for series in a power series field ℝ((t Γ)) with coefficients in ℝ. Subsequently, we define a natural notion of analytic function at a point of ℝ((t Γ))m. Then, given a real function f analytic on a open box I of ℝ m , we extend f to a function f ★ which is analytic on a subset of ℝ((t Γ)) m containing I. We prove that the functions f ★ share with real analytic functions certain basic properties: they are , they have usual Taylor development, they satisfy the inverse function theorem and the implicit function theorem.

Embedding ordered fields in formal power series fields

In the paper it is shown how an embedding of an ordered field F into a formal power series field can be extended canonically to an embedding of any simple extension F( y) of F. Properties of the extended embedding are studied in detail. Several applications are given.

Non-crossed products over function fields

Noncrossed product division algebras are constructed over all function fields and iterated power series fields over global fields, using Hilbert's Irreducibility Theorem and the construction of [B]. Minimum indexes obtained are p2 for odd p and 23 otherwise. Examples are obtained with large index to exponent ratio.

Locally unbounded topological fields with topological nilpotents

We construct some locally unbounded topological fields having topologically nilpotent elements; this answers a question of Heine. The underlying fields are subfields of fields of formal power series. In particular, we get a locally unbounded topological f

Diskret bewertete Ternärkörper und Hahn-Ternärk/ouml;rper auf $ \Bbb Z $

In this paper, we derive criteria for the regular embeddability of a discretely valued ternary field into an appropriate Hahn ternary field of formal power series with coefficients in the residue ternary field and exponents in $ \Bbb Z $ . Furthermore, we discuss these criteria also for richer algebraic structures and we give an example for the division algebra case.

Power Series and p-Adic Algebraic Closures

We describe a presentation of the completion of the algebraic closure of the ring of Witt vectors of an algebraically closed field of characteristic p>0. The construction uses “generalized power series in p” as constructed by Poonen, based on an example of Lampert, and also makes use of an analogous construction of the algebraic closure of a field of power series in positive characteristic.

Elementary properties of power series fields over finite fields

AbstractIn spite of the analogies between ℚp and which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for ℚp, to the case of does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on . We formulate an elementary property expressing this action and show that it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic.

The algebraic closure of the power series field in positive characteristic

For K an algebraically closed field, let K((t)) denote the quotient field of the power series ring over K. The “Newton-Puiseux theorem” states that if K has characteristic 0, the algebraic closure of K((t)) is the union of the fields K((t1/n)) over n ∈ N. We answer a question of Abhyankar by constructing an algebraic closure of K((t)) for any field K of positive characteristic explicitly in terms of certain generalized power series.

Crystals via the affine Grassmannian

Let G be a connected reductive group over ℂ, and let $\mathfrak {g}$∨ be the Langlands dual Lie algebra. Crystals for $\mathfrak {g}$∨ are combinatorial objects that were introduced by M. Kashiwara (cf., e.g., [6]) as certain "combinatorial skeletons" of finite-dimensional representations of $\mathfrak {g}$∨. For every dominant weight λ of $\mathfrak {g}$∨ Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group Uq($\mathfrak {g}$∨) and then specializing it to q=0. Other (independent) constructions of B(λ) were given by G. Lusztig (cf. [9]) using the combinatorics of root systems and by P. Littelmann (cf. [7]) using the "Littelmann path model." It was also shown in [5] that the family of crystals B(λ) is unique if certain reasonable conditions are imposed (cf. Theorem 1.1). The purpose of this paper is to give another (rather simple) construction of the crystals B(λ) using the geometry of the affine Grassmannian \mathscr {G}$G=G($\mathscr{K}$)/G($\mathscr{O}$) of the group G, where $\mathscr{K}$=ℂ((t)) is the field of Laurent power series and $\mathscr{O}$=ℂ[[t]] is the ring of Taylor series. We then check that the family B(λ) satisfies the conditions of the uniqueness theorem from [5], which shows that our crystals coincide with those constructed in the references above. It would be interesting to find these isomorphisms directly (cf., however, [10]).

Sur le théorème de l'indice des équations différentielles p-adiques IV

This paper works out the structure of singular points of p-adic differential equations (i.e. differential modules over the ring of functions analytic in some annulus with external radius 1). Surprisingly results look like in the formal case (differential modules over a one variable power series field) but proofs are much more involved. However, unlike in the Turritin theorem, even after ramification, in the p-adic theory there are irreducible objects of rank >1. The first part is devoted to the definition of p-adic slopes and to a decomposition along p-adic slopes theorem. The case of slope 0 (p-adic analogue of the regular singular case) was already studied in the paper with the same title but number II [Ann. of Math. (2) 146 (1997), 345-410]. The second part states several index existence theorems and index formulas. As a consequence, vertices of the Newton polygon built from p-adic slopes are proved to have integral components (analogue of the Hasse-Arf theorem). After the work of the second author, existence of index implies finitness of p-adic (Monsky-Washnitzer) cohomology for affine varieties over finite fields. The end of the paper outlines the construction of a p-adic-coefficient category over curves (over a finite field) with all needed finitness properties. In the paper with the same title but number IV [Invent. Math. 143 (2001), 629-672], further insights are given.

A Survey of Diophantine Approximationin Fields of Power Series

The fields of power series (or perhaps better called formal numbers) are analogues of the field of real numbers. Many questions in number theory which have been studied in the setting of the real numbers can be transposed to the setting of the power series. The study of rational approximation to algebraic real numbers has been intensively developped starting from the middle of the nineteenth century with the work of Liouville up to the celebrated theorem of Roth established in 1955. In the last thirty years, several mathematicians have studied diophantine approximation in fields of power series. We present here a summary of the present knowledge on this subject, emphasizing the analogies and differences with the situation in the real numbers case.