scholarly journals On the realizable classes of the square root of the inverse different in the unitary class group

2017 ◽  
Vol 13 (04) ◽  
pp. 913-932 ◽  
Author(s):  
Sin Yi Cindy Tsang
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Bilinear Form ◽  
Class Group ◽  
Finite Abelian Group ◽  
Symmetric Bilinear Form ◽  
Square Root ◽  
Ring Of Integers ◽  
Trace Map ◽  
Odd Order

Let [Formula: see text] be a number field with ring of integers [Formula: see text] and let [Formula: see text] be a finite abelian group of odd order. Given a [Formula: see text]-Galois [Formula: see text]-algebra [Formula: see text], write [Formula: see text] for its trace map and [Formula: see text] for its square root of the inverse different, where [Formula: see text] exists by Hilbert’s formula since [Formula: see text] has odd order. The pair [Formula: see text] is locally [Formula: see text]-isometric to [Formula: see text] whenever [Formula: see text] is weakly ramified, in which case it defines a class in the unitary class group [Formula: see text] of [Formula: see text]. Here [Formula: see text] denotes the canonical symmetric bilinear form on [Formula: see text] defined by [Formula: see text] for all [Formula: see text]. We will study the set of all such classes and show that a subset of them forms a subgroup of [Formula: see text].

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.

An arithmetic characterization of algebraic number fields with a given class group

1983 ◽  
Vol 94 (1) ◽  
pp. 23-28 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Class Group ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Unique Factorization ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ring Of Integers

Let R be the ring of integers of a number field K with class group G. It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R. The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewicz's problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.

scholarly journals Computations in Relative Algebraic K-Groups

2009 ◽  
Vol 12 ◽  
pp. 166-194 ◽  
Author(s):  
Werner Bley ◽  
Stephen M. J. Wilson
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Number Field ◽  
Class Group ◽  
Discrete Logarithm ◽  
Torsion Subgroup ◽  
Large Classes ◽  
Ring Of Integers

Let G be finite group and K a number field or a p-adic field with ring of integers OK. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K0(OK[G], K) as an abstract abelian group. We also give algorithms to solve the discrete logarithm problems in K0(OK[G], K) and in the locally free class group cl(OK[G]). All algorithms have been implemented in Magma for the case K = Q.In the second part of the manuscript we prove formulae for the torsion subgroup of K0(Z[G], Q) for large classes of dihedral and quaternion groups.

scholarly journals Picard Groups and Refined Discrete Logarithms

2005 ◽  
Vol 8 ◽  
pp. 1-16 ◽  
Author(s):  
W. Bley ◽  
M. Endres
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Finite Abelian Group ◽  
Discrete Logarithm ◽  
Quadratic Number ◽  
Prime Degree ◽  
Quadratic Number Field ◽  
Picard Groups ◽  
Imaginary Quadratic Number ◽  
Ring Of Algebraic Integers

AbstractLet K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

scholarly journals On Effective Witt Decomposition and the Cartan–Dieudonné Theorem

2007 ◽  
Vol 59 (6) ◽  
pp. 1284-1300 ◽  
Author(s):  
Lenny Fukshansky
Keyword(s):  
Number Field ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Orthogonal Decomposition ◽  
Classical Theorem ◽  
One Dimensional ◽  
Effective Version ◽  
Special Version ◽  
Small Height ◽  
Hyperbolic Planes

AbstractLetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.

scholarly journals On the Steinitz module and capitulation of ideals

2000 ◽  
Vol 160 ◽  
pp. 1-15
Author(s):  
Chandrashekhar Khare ◽  
Dipendra Prasad
Keyword(s):  
Number Field ◽  
Algebraic Curve ◽  
Class Group ◽  
Algebraic Curves ◽  
Finite Extension ◽  
Number Fields ◽  
Line Bundles ◽  
Exterior Power ◽  
Ring Of Integers ◽  
And Function

AbstractLet L be a finite extension of a number field K with ring of integers and respectively. One can consider as a projective module over . The highest exterior power of as an module gives an element of the class group of , called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover Y of an algebraic curve X, the Steinitz module as an element of the Picard group of X is the sum of the line bundles on X which become trivial when pulled back to Y. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in is said to capitulate in L if its extension to is a principal ideal.)

The Kernel of and the 4-Rank of K2(O)

1992 ◽  
Vol 35 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Ruth I. Berger
Keyword(s):  
Number Field ◽  
Upper Bound ◽  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ring Of Integers

AbstractAn upper bound is given for the order of the kernel of the map on Sideal class groups that is induced by For some special types of number fields F the connection between the size of the above kernel for and the units and norms in are examined. Let K2(O) denote the Milnor K-group of the ring of integers of a number field. In some cases a formula by Conner, Hurrelbrink and Kolster is extended to show how closely the 4-rank of is related to the 4-rank of the S-ideal class group of

scholarly journals Bounds for the size of integral solutions to Ym = f(X)

1999 ◽  
Vol 42 (1) ◽  
pp. 127-141
Author(s):  
Dimitrios Poulakis
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Diophantine Equations ◽  
Algebraic Number Field ◽  
Upper Bounds ◽  
Ring Of Integers ◽  
Odd Order ◽  
Integral Solutions

Let K be an algebraic number field with ring of integers OK and f(X) ∈ OK[X]. In this paper we establish improved explicit upper bounds for the size of solutions in OK, of diophantine equations Y2 = f(X), where f(X) has at least three roots of odd order, and Ym = f(X), where m is an integer ≥ 3 and f(X) has at least two roots of order prime to m.

AN ARITHMETICAL INVARIANT OF ORBITS OF AFFINE ACTIONS AND ITS APPLICATION TO SIMILARITY CLASSES OF QUADRATIC SPACES

2010 ◽  
Vol 06 (06) ◽  
pp. 1215-1253
Author(s):  
ANTHONY C. KABLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Number Field ◽  
General Framework ◽  
Class Group ◽  
Regular Function ◽  
Affine Variety ◽  
Natural Parameter ◽  
Ring Of Integers ◽  
Affine Actions

Given an action of an affine algebraic group on an affine variety and a relatively invariant regular function, all defined over the ring of integers of a number field and having suitable additional properties, an invariant of the rational orbits of the action is defined. This invariant, the reduced replete Steinitz class, takes its values in the reduced replete class group of the number field. The general framework is then applied to obtain an invariant of similarity classes of non-degenerate quadratic spaces of even rank. The invariant is related to more familiar invariants. It is shown that if the similarity classes are weighted by the volume of an associated automorphism group then their reduced replete Steinitz classes are asymptotically uniformly distributed with respect to a natural parameter.

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