Hilbert–Speiser number fields for a prime p inside the p-cyclotomic field

A lattice construction using Z-submodules of rings of integers of number ﬁelds is presented. The construction yields rotated versions of the laminated lattices A_n for n = 2,3,4,5,6, which are the densest lattices in their respective dimensions. The sphere packing density of a lattice is a function of its packing radius, which in turn can be directly calculated from the minimum squared Euclidean norm of the lattice. Norms in a lattice that is realized by a totally real number ﬁeld can be calculated by the trace form of the ﬁeld restricted to its ring of integers. Thus, in the present work, we also present the trace form of the maximal real subﬁeld of a cyclotomic ﬁeld. Our focus is on totally real number ﬁelds since their associated lattices have full diversity. Along with high packing density, the full diversity feature is desirable in lattices that are used for signal transmission over both Gaussian and Rayleigh fading channels.

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A Note on Ideal Class Groups

Nagoya Mathematical Journal ◽

10.1017/s0027763000012046 ◽

1966 ◽

Vol 27 (1) ◽

pp. 239-247 ◽

Cited By ~ 9

Author(s):

Kenkichi Iwasawa

Keyword(s):

Class Group ◽

Cyclotomic Field ◽

Theoretical Method ◽

Number Fields ◽

Ideal Class ◽

Ideal Class Group ◽

Class Groups ◽

Algebraic Number Fields ◽

Ideal Class Groups ◽

The Ideal

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

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Generalized Beilinson Elements and Generalized Soulé Characters

Canadian Journal of Mathematics ◽

10.4153/s0008414x20000073 ◽

2020 ◽

pp. 1-30

Author(s):

Kenji Sakugawa

Keyword(s):

Arbitrary Number ◽

Cyclotomic Field ◽

Number Fields ◽

Field Case

Abstract The generalized Soulé character was introduced by H. Nakamura and Z. Wojtkowiak and is a generalization of Soulé’s cyclotomic character. In this paper, we prove that certain linear sums of generalized Soulé characters essentially coincide with the image of generalized Beilinson elements in K-groups under Soulé’s higher regulator maps. This result generalizes Huber–Wildeshaus’ theorem, which is a cyclotomic field case of our results, to an arbitrary number fields.

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A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field Q(z_p)

TEMA (São Carlos) ◽

10.5540/tema.2019.020.03.561 ◽

2019 ◽

Vol 20 (3) ◽

pp. 561

Author(s):

Antonio A. Andrade ◽

Everton L. Oliveira ◽

José C. Interlando

Keyword(s):

Fading Channel ◽

Rayleigh Fading ◽

Cyclotomic Field ◽

Number Fields ◽

Rayleigh Fading Channel ◽

Ring Of Integers ◽

Full Diversity ◽

Alternative Approach ◽

Totally Real ◽

Modulation Diversity

The theory of lattices have shown to be useful in information theory and rotated lattices with high modulations diversity have been extensively studied as an alternative approach for transmission over a Rayleigh-fading channel, where the performance of this modulation schemes essentially depends of the modulation diversity and of the minimum product distance to achieve substantial coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminant. In this paper, we present a construction of rotated lattice for the Rayleigh fading channel in Euclidean spaces with full diversity, where this construction is through a totally real subfield K of the cyclotomic field Q(z_p), where p is an odd prime, obtained by endowing their ring of integers.

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REPRESENTING Ω(l∞) FOR REAL ABELIAN FIELDS

Journal of Algebra and Its Applications ◽

10.1142/s0219498803000404 ◽

2003 ◽

Vol 02 (03) ◽

pp. 237-276 ◽

Cited By ~ 4

Author(s):

JÜRGEN RITTER ◽

ALFRED WEISS

Keyword(s):

Real Number ◽

Prime Number ◽

Cyclotomic Field ◽

Number Fields ◽

Iwasawa Theory ◽

Root Number ◽

Abelian Extensions ◽

Main Conjecture ◽

Abelian Fields ◽

Totally Real

For real subfields K of a cyclotomic field ℚ(ς) we remove the tameness assumption at a given odd prime number l, which was needed in [11] in order to establish the equivalence of the Lifted Root Number Conjecture at l and an equivariant main conjecture of Iwasawa theory for abelian extensions of totally real number fields K/k.

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Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.1.76-90 ◽

2021 ◽

Vol 27 (1) ◽

pp. 76-90

Author(s):

Lukasz Nizio ◽

Keyword(s):

Finite Number ◽

Cyclotomic Field ◽

Number Fields ◽

Lattice Points ◽

Quadratic Fields ◽

Quadratic Number ◽

Imaginary Quadratic Fields ◽

Integral Points ◽

Imaginary Quadratic Number ◽

Ring Of Algebraic Integers

We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite number of \alpha-lattice points for a fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring of algebraic integers of \mathbb{K}. These varieties arise from equations of the form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), where F is a rational function, g and r are polynomials over \mathbb{K}, and the degree of r is relatively small. We also give an example of an affine variety of dimension two, with a finite number of algebraic integral points. This variety is defined over the cyclotomic field \mathbb{Q}(\xi_3)=\mathbb{Q}(\sqrt{-3}).

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