Ring of integers in an extension of prime degree of a local field as a Galois module

1976 ◽  
Vol 6 (3) ◽  
pp. 227-238
Author(s):  
Z. I. Borevich ◽  
S. V. Vostokov
Keyword(s):  
Local Field ◽  
Galois Module ◽  
Prime Degree ◽  
Ring Of Integers

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

Trace form associated to cyclic number fields of ramified odd prime degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050080
Author(s):  
Robson R. Araujo ◽  
Ana C. M. M. Chagas ◽  
Antonio A. Andrade ◽  
Trajano P. Nóbrega Neto
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Trace Form ◽  
Prime Degree ◽  
Ring Of Integers ◽  
Algebraic Lattices ◽  
Center Density

In this work, we computate the trace form [Formula: see text] associated to a cyclic number field [Formula: see text] of odd prime degree [Formula: see text], where [Formula: see text] ramified in [Formula: see text] and [Formula: see text] belongs to the ring of integers of [Formula: see text]. Furthermore, we use this trace form to calculate the expression of the center density of algebraic lattices constructed via the Minkowski embedding from some ideals in the ring of integers of [Formula: see text].

On the self-duality of a ring of integers as a Galois module

1978 ◽  
Vol 46 (2) ◽  
pp. 173-177 ◽  
Author(s):  
M. J. Taylor
Keyword(s):  
The Self ◽  
Galois Module ◽  
Ring Of Integers ◽  
Self Duality

Relative Galois module structure of rings of integers and elliptic functions

1983 ◽  
Vol 94 (3) ◽  
pp. 389-397 ◽  
Author(s):  
M. J. Taylor
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Elliptic Functions ◽  
Complex Numbers ◽  
Module Structure ◽  
Galois Module ◽  
Integral Ideal ◽  
Ring Of Integers ◽  
Rings Of Integers ◽  
Imaginary Number

Let K be a quadratic imaginary number field with discriminant less than −4. For N either a number field or a finite extension of the p-adic field p, we let N denote the ring of integers of N. Moreover, if N is a number field then we write for the integral closure of [½] in N. For an integral ideal & of K we denote the ray classfield of K with conductor & by K(&). Once and for all we fix a choice of embedding of K into the complex numbers .

scholarly journals Compatible systems and ramification

2019 ◽  
Vol 155 (12) ◽  
pp. 2334-2353 ◽  
Author(s):  
Qing Lu ◽  
Weizhe Zheng
Keyword(s):  
Finite Field ◽  
Finite Type ◽  
Local Field ◽  
Ring Of Integers

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

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