Definability of the ring of integers in pro-p galois extensions of number fields

2000 ◽  
Vol 118 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Carlos R. Videla
Keyword(s):  
Number Fields ◽  
Galois Extensions ◽  
Ring Of Integers

scholarly journals Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

2020 ◽  
Author(s):  
Chris Bruce
Keyword(s):  
Phase Transitions ◽  
Class Group ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
C Algebras ◽  
Rings Of Algebraic Integers ◽  
Beta 2 ◽  
Factor State ◽  
Type Classification

Abstract We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta \in [1,2]$, there is a unique KMS$_\beta $ state, and we prove that it is a factor state of type III$_1$. There are phase transitions at $\beta =2$ and $\beta =\infty $ involving a quotient of a ray class group. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

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].

scholarly journals There are no universal ternary quadratic forms over biquadratic fields

2020 ◽  
Vol 63 (3) ◽  
pp. 861-912 ◽  
Author(s):  
Jakub Krásenský ◽  
Magdaléna Tinková ◽  
Kristýna Zemková
Keyword(s):  
Quadratic Forms ◽  
Number Fields ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Mild Conditions ◽  
Positive Unit ◽  
Positive Elements ◽  
Totally Positive ◽  
Biquadratic Fields ◽  
Totally Real

AbstractWe study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

scholarly journals LOCAL HEIGHTS ON ELLIPTIC CURVES AND INTERSECTION MULTIPLICITIES

2012 ◽  
Vol 08 (06) ◽  
pp. 1477-1484
Author(s):  
VINCENZ BUSCH ◽  
JAN STEFFEN MÜLLER
Keyword(s):  
Elliptic Curves ◽  
Short Note ◽  
Number Fields ◽  
Intersection Theory ◽  
Ring Of Integers ◽  
Regular Model ◽  
Intersection Multiplicities

In this short note we prove a formula for local heights on elliptic curves over number fields in terms of intersection theory on a regular model over the ring of integers.

scholarly journals Geometric-progression-free sets over quadratic number fields

2017 ◽  
Vol 147 (2) ◽  
pp. 245-262
Author(s):  
Andrew Best ◽  
Karen Huan ◽  
Nathan McNew ◽  
Steven J. Miller ◽  
Jasmine Powell ◽  
...  
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Geometric Progression ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Quadratic Number Fields ◽  
Geometric Progressions ◽  
Upper Density ◽  
Free Sets

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

A Stickelberger Condition on Cyclic Galois Extensions

1982 ◽  
Vol 34 (3) ◽  
pp. 686-690 ◽  
Author(s):  
L. N. Childs
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Galois Extension ◽  
Class Group ◽  
Finite Abelian Group ◽  
Algebraic Number Field ◽  
Primitive Elements ◽  
Galois Extensions ◽  
Ring Of Integers ◽  
Isomorphism Classes

LetRbe a commutative ring,Ca finite abelian group,Sa Galois extension ofRwith groupC, in the sense of [1]. ViewingSas anRC-module defines the Picard invariant map [4] from the Harrison group Gal (R,C) of isomorphism classes of Galois extensions ofRwith groupCto CI (RC), the class group ofRC. The image of the Picard invariant map is known to be contained in the subgrouphCl (RC) of primitive elements of CI (RC) (for definition see below). Characterizing the image of the Picard invariant map has been of some interest, for the image describes the extent of failure of Galois extensions to have normal bases.LetRbe the ring of integers of an algebraic number fieldK.

Sign in / Sign up

Export Citation Format

Share Document