Determination of the Number of Ideal Classes in the Cyclotomic Field of the m-th Roots of Unity

1998 ◽  
pp. 207-215
Author(s):  
David Hilbert
Keyword(s):  
Cyclotomic Field ◽  
Roots Of Unity

Fermat Jacobians of Prime Degree over Finite Fields

1999 ◽  
Vol 42 (1) ◽  
pp. 78-86 ◽  
Author(s):  
Josep González
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Cyclotomic Field ◽  
Roots Of Unity ◽  
Characteristic P ◽  
Prime Degree ◽  
Numerical Criterion

AbstractWe study the splitting of Fermat Jacobians of prime degree l over an algebraic closure of a finite field of characteristic p not equal to l. We prove that their decomposition is determined by the residue degree of p in the cyclotomic field of the l-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

Galois Actions on the Eigenproblem of the Heisenberg Heptagon

2012 ◽  
Vol 19 (02) ◽  
pp. 1250012 ◽  
Author(s):  
Jan Milewski ◽  
Grzegorz Banaszak ◽  
Tadeusz Lulek ◽  
Mirosław Łabuz ◽  
Ryszard Stagraczyński
Keyword(s):  
Fourier Transforms ◽  
Cyclotomic Field ◽  
Galois Groups ◽  
Point Of View ◽  
Kummer Theory ◽  
Computer Based ◽  
Heisenberg Field ◽  
Galois Actions ◽  
Xxx Model

We analyse the exact solution of the eigenproblem for the Heisenberg Hamiltonian of magnetic heptagon, i.e. the ring of N = 7 nodes, each with spin 1/2, within the XXX model with nearest neighbour interactions, from the point of view of finite extensions of the field ℚ of rationals. We point out, as the main result, that the associated arithmetic structure of these extensions makes natural an introduction of some Galois qubits. They are two-dimensional subspaces of the Hilbert space of the model, which admit a quantum informatic interpretation as elementary memory units for a (hypothetical) computer, based on their distinctive properties with respect to the action of related Galois group for indecomposable factors of the secular determinant. These Galois qubits are nested on the lattice of subfields which involves several minimal fields for determination of eigenstates (the complex Heisenberg field), spectrum (the real Heisenberg field), and Fourier transforms of magnetic configurations (the cyclotomic field, based on the simple 7th root of unity). The structure of the corresponding lattice of Galois groups is presented in terms of Kummer theory, and its physical interpretation is indicated in terms of appropriate permutations of eigenstates, energies, and density matrices.

scholarly journals ORIGAMI RINGS

2012 ◽  
Vol 92 (3) ◽  
pp. 299-311 ◽  
Author(s):  
JOE BUHLER ◽  
STEVE BUTLER ◽  
WARWICK DE LAUNEY ◽  
RON GRAHAM
Keyword(s):  
Complex Plane ◽  
Cyclotomic Field ◽  
Roots Of Unity ◽  
Precise Description ◽  
Infinite Sets ◽  
Set Of Points

AbstractMotivated by mathematical aspects of origami, Erik Demaine asked which points in the plane can be constructed by using lines whose angles are multiples of $\pi /n$ for some fixed $n$. This has been answered for some specific small values of $n$ including $n=3,4,5,6,8,10,12,24$. We answer this question for arbitrary $n$. The set of points is a subring of the complex plane $\mathbf {C}$, lying inside the cyclotomic field of $n$th roots of unity; the precise description of the ring depends on whether $n$is prime or composite. The techniques apply in more general situations, for example, infinite sets of angles, or more general constructions of subsets of the plane.

Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials

1991 ◽  
Vol 43 (5) ◽  
pp. 1055-1064 ◽  
Author(s):  
Walter Tautz ◽  
Jaap Top ◽  
Alain Verberkmoes
Keyword(s):  
Cyclotomic Field ◽  
Prime Numbers ◽  
Explicit Construction ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
The Other ◽  
Roots Of Unity ◽  
Permutation Polynomials ◽  
Endomorphism Algebra ◽  
Real Multiplication

AbstractThe aim of this paper is to present a very explicit construction of one parameter families of hyperelliptic curves C of genus (p−1 )/ 2, for any odd prime number p, with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q(2 cos(2π/p)) of the cyclotomic field Q(e2π i/p).Two proofs of the fact that the constructed curves have this property will be given. One is by providing a double cover with the pth roots of unity in its automorphism group. The other is by explicitly writing down equations of a correspondence in C x C which defines multiplication by 2cos(2π/ p) on the jacobian of C. As a byproduct we obtain polynomials which define bijective maps Fℓ → Fℓ for all prime numbers in certain congruence classes.

scholarly journals MODULI INTERPRETATION OF EISENSTEIN SERIES

2012 ◽  
Vol 08 (03) ◽  
pp. 715-748 ◽  
Author(s):  
KAMAL KHURI-MAKDISI
Keyword(s):  
Elliptic Curve ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Cyclotomic Field ◽  
Hecke Operators ◽  
Roots Of Unity ◽  
Graded Algebra ◽  
Systematic Method ◽  
Modular Curve ◽  
Exact Arithmetic

Let ℓ ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(ℓ) over any field k where 6ℓ is invertible and k contains the ℓth roots of unity. These forms generate a graded algebra [Formula: see text], which, over C, is generated by the Eisenstein series of weight 1 on Γ(ℓ). The main result of this article is that, when k = C, the ring [Formula: see text] contains all modular forms on Γ(ℓ) in weights ≥ 2. The proof combines algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions. Our results give a systematic method to produce models for the modular curve X(ℓ) defined over the ℓth cyclotomic field, using only exact arithmetic in the ℓ-torsion field of a single Q-rational elliptic curve E0.

Galactic orbits of stars

1966 ◽  
Vol 25 ◽  
pp. 93-97
Author(s):  
Richard Woolley
Keyword(s):  
Globular Cluster ◽  
Potential Field ◽  
High Velocity ◽  
Velocity Vector ◽  
Variable Stars ◽  
The Sun ◽  
Proper Motions ◽  
Rr Lyrae ◽  
The Galaxy

It is now possible to determine proper motions of high-velocity objects in such a way as to obtain with some accuracy the velocity vector relevant to the Sun. If a potential field of the Galaxy is assumed, one can compute an actual orbit. A determination of the velocity of the globular clusterωCentauri has recently been completed at Greenwich, and it is found that the orbit is strongly retrograde in the Galaxy. Similar calculations may be made, though with less certainty, in the case of RR Lyrae variable stars.

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