Extensions Of Local Fields Research Articles

Extensions of local fields given by 3-term Eisenstein polynomials

Let K be a local field with residue characteristic p and let L/K be a totally ramified extension of degree pk. In this paper we show that if L/K has only two distinct indices of inseparability then there exists a uniformizer πL for L whose minimum polynomial over K has at most three terms. This leads to an explicit classification of extensions with two indices of inseparability. Our classification extends work of Amano, who considered the case k=1.

Calculations in the Generalized Lubin–Tate Theory

In this paper, various extensions of local fields are considered. For arbitrary finite extension K of the field of p-adic numbers, the maximum Abelian extension KAb/K and the corresponding Galois group can be described using the well-known Lubin–Tate theory. It is represented as a direct product of groups obtained using the maximum unramified extension of K and a fully ramified extension obtained using the roots of some endomorphisms of Lubin–Tate formal groups. We consider the so-called “generalized Lubin–Tate formal groups” and extensions obtained by adding the roots of their endomorphisms to the field under consideration. Using the fact that a correctly chosen generalized formal group coincides with the classical one over unramified finite extension Tm of degree m of field K, it was possible to obtain the Galois group of the extension (Tm)Ab/K. The main result of the work, is an explicit description of the Galois group of the extension (Kur)Ab/K, where Kur is the maximum unramified extension of K. Similar methods are also used to study ramified extensions of the field K.

Cohomology of Formal Modules over Local Fields

The structure of the first Galois cohomology groups for the group of points of a formal module in extensions of local fields is studied. A complete description for unramified extensions and classical formal group laws is obtained.

On the Reduced Group of Principal Units in Cyclic Extensions of Local Fields

The reduced group of principal units of cyclic extensions of local fields is considered as a Galois module. This module is studied by examining the Jordan canonical form of the generating automorphism of the Galois group.

Variation of Tamagawa numbers of semistable abelian varieties in field extensions

AbstractWe investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on thep-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the formy2=f(x), under some simplifying hypotheses.

Extensions of local fields and elementary symmetric polynomials

Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension of degree $n=up^{\nu}$. Let $\sigma_1,\dots,\sigma_n$ denote the $K$-embeddings of $L$ into a separable closure $K^{sep}$ of $K$. For $1\le h\le n$ let $e_h(X_1,\dots,X_n)$ denote the $h$th elementary symmetric polynomial in $n$ variables, and for $\alpha\in L$ set $E_h(\alpha) =e_h(\sigma_1(\alpha),\dots,\sigma_n(\alpha))$. Set $j=\min\{v_p(h),\nu\}$. We show that for $r\in\mathbb{Z}$ we have $E_h(\mathcal{M}_L^r)\subset \mathcal{M}_K^{\lceil(i_j+hr)/n\rceil}$, where $i_j$ is the $j$th index of inseparability of $L/K$. In certain cases we also show that $E_h(\mathcal{M}_L^r)$ is not contained in any higher power of $\mathcal{M}_K$.

Ramification filtrations of certain abelian Lie extensions of local fields

Let G⊂xFq〚x〛 (q is a power of the prime p) be a subset of formal power series over a finite field such that it forms a compact abelian p-adic Lie group of dimension d≥1. We establish a necessary and sufficient condition for the APF extension of local field corresponding to (Fq⸨x⸩,G) under the field of norms functor to be an extension of p-adic fields. We then apply this result to study invertible power series over a ring of p-adic integers which commute with a fixed noninvertible power series under composition.

Explicit local reciprocity for tame extensions

AbstractWe consider a tamely ramified abelian extension of local fields of degreen, without assuming the presence of thenth roots of unity in the base field. We give an explicit formula which computes the local reciprocity map in this situation.

Tower of the maximal abelian extensions of local fields and its application

In this paper, we study the algebraic structure of principal units in the tower of the maximal abelian extensions of local fields of characteristic zero and the corresponding Galois groups at each level. As an application, we show the finiteness result for the number of coverings with a given degree of the maximal abelian extension of a local field in characteristic zero. The number of p-coverings for \({\mathbb{Q}_p}\) is computed explicitly.

The Compositum of Wild Extensions of Local Fields of Prime Degree

In this paper we present a general view of the totally and wildly ramified extensions of degree p of a p-adic field K. Our method consists in deducing the properties of the set of all extensions of degree p of K from the study of the compositum ${\cal C}_K(p)$ of all its elements. We show that in fact ${\cal C}_K(p)$ is the maximal abelian extension of exponent p of F = F(K), where F is the compositum of all cyclic extensions of K of degree dividing p − 1. By our method, it is fairly simple to recover the distribution of the extensions of K of degree p (and also of their isomorphism classes) according to their discriminant.

A formula for Tignol's constant

Let ( K , v ) be a Henselian valued field and ( L , w ) be a finite separable extension of ( K , v ) . In 2004, it was proved that the set A L / K defined by A L / K = { v ( Tr L / K ( α ) ) − w ( α ) | α ∈ L , α ≠ 0 } has a minimum element if and only if [ L : K ] = e f where e , f are the ramification index and the residual degree of w / v , i.e., ( L , w ) / ( K , v ) is defectless. The constant min A L / K was first introduced by Tignol and is referred to as Tignol's constant. In 2005, K. Ota and Khanduja gave a formula for min A L / K when ( L , w ) / ( K , v ) is an extension of local fields. In this paper, we give this formula when ( L , w ) is any finite separable defectless extension of a Henselian valued field of arbitrary rank and thereby generalize some well-known results of Dedekind regarding “different” of extensions of algebraic number fields and ramification of prime ideals.

Extensions of local fields and truncated power series

Let K be a finite tamely ramified extension of Q p and let L / K be a totally ramified ( Z / p n Z ) -extension. Let π L be a uniformizer for L, let σ be a generator for Gal ( L / K ) , and let f ( X ) be an element of O K [ X ] such that σ ( π L ) = f ( π L ) . We show that the reduction of f ( X ) modulo the maximal ideal of O K determines a certain subextension of L / K up to isomorphism. We use this result to study the field extensions generated by periodic points of a p-adic dynamical system.

Galois Module Structure of the Integers in Wildly Ramified Cp × Cp Extensions

AbstractLet L/K be a finite Galois extension of local fields which are finite extensions of ℚp, the field of p-adic numbers. Let Gal(L/K) = G, and 𝔒L and ℤp be the rings of integers in L and ℚp, respectively. And let 𝔓L denote the maximal ideal of 𝔒L. We determine, explicitly in terms of specific indecomposable ℤp[G]-modules, the ℤp[G]-module structure of 𝔒L and 𝔓L, for L, a composite of two arithmetically disjoint, ramified cyclic extensions of K, one of which is only weakly ramified in the sense of Erez [6].

Konstruktion lokalkompakter Fastk�rper

It is the aim of this paper to present an extensive class of disconnected locally compact nearfields. They are all Dickson nearfields and derived from tamely ramified extensions of local fields.

Quadratic subfields on quartic extensions of local fields

We show that any quartic extension of a local field of odd residue characteristic must contain an intermediate field. A consequence of this is that local fields of odd residue characteristic do not have extensions with Galois group A4 or S4. Counterexamples are given for even residue characteristic.

On the extensions of local fields generated by the torsion points of some formal groups

On the extensions of local fields generated by the torsion points of some formal groups

Adams' operation and the symbol for the normed residue

We study the possibility of a meaningful extension of the ring of cohomologies of a point using algebraic extensions of local fields. It appears that complex K-theory is a good model problem for this. We prove that the effect of the Adams operation on the cohomologies of a point in extended K-theory coincides with the symbol for the local Artin reciprocity law. K-theories contained in a fixed K-theory with extended ring of cohomologies of a point are defined by higher branching groups.

On p-Regular Extensions of Local Fields

On p-Regular Extensions of Local Fields