A characterization of complete subfields of a complete valuated field (original) (raw)
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Journal of Number Theory, 1998
The so called``p-adic analog of the field of complex numbers'' C p (see [6, 7, 21]) seems to be a very interesting object to study, both from an algebraic and an analytic point of view. Particularly interesting are its closed subfields. A first account on them can be find in [7] (see also [14] and especially [21]). Here we try to give some new aspects and results on the elements of C p which are transcendental over Q p. These elements will be called simplỳ`t ranscendental''. Our paper has six sections. In the first section we recall basic results, definitions and notations. In the second one we define``distinguished sequences'' and prove that they permit to construct transcendental elements and also to associate to any transcendental element an infinite set of numerical invariants. Although the invariants of a given transcendental element do not define it uniquely, they tell much about it. In Section 3 we consider the conjugate class (or orbit) of a transcendental element. This orbit is compact and totally disconnected. In Theorem 3.7 we give an analytic criterion to determine the conjugates of a transcendental element. In Section 4 we deal with the so-called generic transcendental elements. Let K be a closed subfield of C p , infinite over Q p. According to [13], there exists a generic transcendental element t of K, i.e. such that K is the topological closure of Q p (t). Starting with a distinguished sequence [: n ] n which defines t we can describe the action of v on K, the residue field and value group of K and the algebraic closure of Q p in K (Theorems 4.1 and 4.2). In Section 5 we introduce a particular kind of generic transcendental elements. Although Theorem 5.4 gives a criterion for two such elements to Article No. NT972198 131 0022-314XÂ98 25.00
The model theory of separably tame valued fields
Journal of Algebra, 2016
A henselian valued field K is called separably tame if its separable-algebraic closure K sep is a tame extension, that is, the ramification field of the normal extension K sep |K is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields.
On fields with the Property (B)
HAL (Le Centre pour la Communication Scientifique Directe), 2014
Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent. We show that L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L * (outside the set of roots of unity) is bounded from below by an absolute constant. We discuss some feature of Property (B): stability by algebraic extensions, relations with field arithmetic. As a as a side result, we prove that the Galois group over Q of the compositum of all totally real fields is torsion free.
Springer undergraduate mathematics series, 2018
Separable Extensions As we already noted in the previous chapter, our final goal is Galois extensions. The last property of field extensions which we need is separability. This property is rather common, for example, all extensions of fields of characteristic zero (thus, in particular, all number fields) have this property. All finite extensions of finite fields also have this property. This is a reason why there are sometimes "simplified presentations" of Galois theory in which one studies only fields of characteristic zero and finite fields. In that case it is not necessary to mention separability and the theoretical background necessary for the main theorems of Galois theory is more modest. We choose to discuss separability, since there are many branches of mathematics in which non-separable field extensions are important. However, in the sequel, those who wish to may disregard the word "separable", accepting that the results are formulated for fields of characteristic zero or finite fields. In fact, Theorem T.8.1 says that all extensions of fields of characteristic 0 or algebraic extensions of finite fields are separable. In this chapter, we characterize separable extensions and prove a well-known "theorem on primitive elements" which says that a finite separable extension can be generated over its ground field by only one element. Since in this book the exercises not related to the theorem on primitive elements are often of a theoretical nature, we choose to present them with more detailed solutions (in order to not oversize the role of this chapter). A polynomial f 2 KOEX is called separable over K if it has no multiple zeros (in any extension L Ã K-see T.5.3). We say that˛2 L Ã K is a separable element over K if it is algebraic and its minimal polynomial over K is separable. We say that L Ã K is a separable extension if every element of L is separable over K. A field K is called perfect if every algebraic extension of it is separable (that is, every irreducible polynomial in KOEX is separable).
Rational separability over a global field
Annals of Pure and Applied Logic, 1996
Let F be a finitely generated field and let j:F -+ N be a weak presentation of F, i.e. an isomorphism from F onto a field whose universe is a subset of N and such that all the field operations are extendible to total recursive functions. Then if RI and R2 are recursive subrings of F, for all weak presentations j of F, j(R,) is Turing reducible to j (R,) if and only if there exists a finite collection of non-constant rational functions {Gi} over F such that for every x E RI for
On common extensions of valued fields
arXiv: Commutative Algebra, 2020
Given a valuation vvv on a field KKK, an extension barv\bar{v}barv to an algebraic closure and an extension www to K(X)K(X)K(X). We want to study the common extensions of barv\bar{v}barv and www to barK(X)\bar{K}(X)barK(X). First we give a detailed link between the minimal pairs notion and the key polynomials notion. Then we prove that in the case when www is a transcendental extension, then any sequence of key polynomials admits a maximal element, and in case this sequence does not contain a limit key polynomial, then any root of the last key polynomial, describe a common extension.
Proceedings of the American Mathematical Society, 2014
Let K K be a number field and let L / K L/K be an infinite Galois extension with Galois group G G . Let us assume that G / Z ( G ) G/Z(G) has finite exponent. We show that L L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L ∗ L^* is bounded from below outside the set of roots of unity by an absolute constant. We also discuss some features of Property (B): stability by algebraic extensions and relations with field arithmetic. As a side result, we prove that the Galois group over Q \mathbb {Q} of the compositum of all totally real fields is torsion free.
Jónssonω0-generated algebraic field extensions
Pacific Journal of Mathematics, 1987
A field K algebraic over its subfield F is said to be a /-extension (for Jόnsson ω o-generated extension) of F if K/F is not finitely generated, but E/F is finitely generated for each proper intermediate field E. We seek to determine the structure of a given /-extension and to determine the class of fields that admit a /-extension. Consideration of Galois /-extensions plays a special role in each of these problems. In §2, we show that a Galois extension K/F is a /-extension if and only if Gύ(K/F) a lim Z/p n Z for some prime p. In §3, we show that F admits a /-extension if the algebraic closure of F is infinite over F-that is, F is neither algebraically closed nor real closed.
Valuations in algebraic field extensions
Journal of Algebra, 2007
Let K → L be an algebraic field extension and ν a valuation of K. The purpose of this paper is to describe the totality of extensions {ν ′ } of ν to L using a refined version of MacLane's key polynomials. In the basic case when L is a finite separable extension and rk ν = 1, we give an explicit description of the limit key polynomials (which can be viewed as a generalization of the Artin-Schreier polynomials). We also give a realistic upper bound on the order type of the set of key polynomials. Namely, we show that if char K = 0 then the set of key polynomials has order type at most N, while in the case char K = p > 0 this order type is bounded above by log p n + 1 ω, where n = [L : K]. Our results provide a new point of view of the the well known formula s j=1 e j f j d j = n and the notion of defect.