scholarly journals Complex Roots of Unity and Normal Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Jean Marie De Koninck ◽  
Imre Kátai
Keyword(s):  
Prime Number ◽  
Roots Of Unity ◽  
Normal Numbers ◽  
Famous Result ◽  
Dirichlet Characters ◽  
Complex Roots ◽  
Selection Of

Given an arbitrary prime number q, set ξ=e2πi/q. We use a clever selection of the values of ξα, α=1,2,…, in order to create normal numbers. We also use a famous result of André Weil concerning Dirichlet characters to construct a family of normal numbers.

AN ELEMENTARY PROOF OF THE CONDUCTOR-DISCRIMINANT FORMULA

2010 ◽  
Vol 06 (05) ◽  
pp. 1191-1197
Author(s):  
GABRIEL VILLA-SALVADOR
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Prime Number ◽  
Elementary Proof ◽  
Abelian Extension ◽  
Roots Of Unity ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
Dirichlet Characters ◽  
Ramification Index

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pn-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.

scholarly journals Roots of Elliptic Scator Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 321
Author(s):  
Manuel Fernandez-Guasti
Keyword(s):  
Abelian Group ◽  
Geometric Interpretation ◽  
Roots Of Unity ◽  
Complex Roots ◽  
De Moivre’S Formula

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

scholarly journals Characterisations of Galois extensions of prime cubed degree

1997 ◽  
Vol 55 (1) ◽  
pp. 99-112 ◽  
Author(s):  
James E. Carter
Keyword(s):  
Prime Number ◽  
Roots Of Unity ◽  
Galois Extensions

Let p be a prime number and let k be a field of characteristic not equal to p. Assuming k contains the appropriate roots of unity, we characterise the non-cyclic Galois extensions of k of degree p3. Concrete examples of such extensions are given for each possible case which can occur, up to isomorphism.

scholarly journals K-theoretic exceptional collections at roots of unity

2010 ◽  
Vol 7 (1) ◽  
pp. 169-201 ◽  
Author(s):  
A. Polishchuk
Keyword(s):  
Prime Number ◽  
Derived Categories ◽  
Torus Action ◽  
Roots Of Unity ◽  
Coherent Sheaves ◽  
Discrete Invariants ◽  
Exceptional Object

AbstractUsing cyclotomic specializations of equivariant K-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if , where the ni's are powers of a fixed prime number p, then the rank of an exceptional object on X is congruent to ±1 modulo p.

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