scholarly journals Monogenity of totally real algebraic extension fields over a cyclotomic field

2016 ◽  
Vol 158 ◽  
pp. 348-355 ◽  
Author(s):  
Nadia Khan ◽  
Shin-ichi Katayama ◽  
Toru Nakahara ◽  
Tsuyoshi Uehara
Keyword(s):  
Cyclotomic Field ◽  
Algebraic Extension ◽  
Totally Real

scholarly journals Constructions of Dense Lattices of Full Diversity

2020 ◽  
Vol 21 (2) ◽  
pp. 299
Author(s):  
A. A. Andrade ◽  
A. J. Ferrari ◽  
J. C. Interlando ◽  
R. R. Araujo
Keyword(s):  
Real Number ◽  
Fading Channels ◽  
Packing Density ◽  
Signal Transmission ◽  
Cyclotomic Field ◽  
Sphere Packing ◽  
Number Fields ◽  
Trace Form ◽  
Full Diversity ◽  
Totally Real

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.

scholarly journals Rotated Z^n-Lattices via Real Subfields of Q(\zeta_2r)

2019 ◽  
Vol 20 (3) ◽  
pp. 445
Author(s):  
Antonio A. Andrade ◽  
José C. Interlando
Keyword(s):  
Cyclotomic Field ◽  
Full Diversity ◽  
Totally Real

A method for constructing rotated Z^n-lattices, with n a power of 2, based on totally real subfields of the cyclotomic field Q(\zeta_{2^r}), where r\geq 4 is an integer, is presented. Lattices exhibiting full diversity in some dimensions n not previously addressed are obtained.

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

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.

REPRESENTING Ω(l∞) FOR REAL ABELIAN FIELDS

2003 ◽  
Vol 02 (03) ◽  
pp. 237-276 ◽  
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.

scholarly journals On number fields with given ramification

2007 ◽  
Vol 143 (6) ◽  
pp. 1359-1373 ◽  
Author(s):  
Gaëtan Chenevier
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Number Fields ◽  
Algebraic Extension ◽  
Natural Maps ◽  
Finite Set ◽  
Totally Real

AbstractLet E be a CM number field and let S be a finite set of primes of E containing the primes dividing a given prime number l and another prime u split above the maximal totally real subfield of E. If ES denotes a maximal algebraic extension of E which is unramified outside S, we show that the natural maps $\mathrm {Gal}(\overline {E_u}/E_u) \longrightarrow \mathrm {Gal}(E_S/E)$ are injective. We discuss generalizations of this result.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6449-6459 ◽  
Author(s):  
Akram Ali ◽  
Siraj Uddin ◽  
Wan Othman ◽  
Cenap Ozel
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Locally Conformal ◽  
Totally Real ◽  
Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

scholarly journals DIOPHANTINE SETS OF POLYNOMIALS OVER ALGEBRAIC EXTENSIONS OF THE RATIONALS

2014 ◽  
Vol 79 (3) ◽  
pp. 733-747
Author(s):  
CLAUDIA DEGROOTE ◽  
JEROEN DEMEYER
Keyword(s):  
Minimal Polynomial ◽  
Finite Extension ◽  
Algebraic Extension ◽  
Recursively Enumerable

AbstractLet L be a recursive algebraic extension of ℚ. Assume that, given α ∈ L, we can compute the roots in L of its minimal polynomial over ℚ and we can determine which roots are Aut(L)-conjugate to α. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of α, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in ℝ, or in a finite extension of ℚp (with p an odd prime). Then we show that subsets of L[X]k that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X].

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

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