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.

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 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.

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.

scholarly journals Équation de Fermat et nombres premiers inertes

2015 ◽  
Vol 11 (08) ◽  
pp. 2341-2351
Author(s):  
Alain Kraus
Keyword(s):  
Finite Number ◽  
Prime Number ◽  
Number Field ◽  
Class Number ◽  
Quadratic Field ◽  
Roots Of Unity ◽  
Mots Clés ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
De Formula

Soient K un corps de nombres et p un nombre premier ≥ 5. Notons μp le groupe des racines p-ièmes de l'unité. On définit p comme étant K-régulier si p ne divise pas le nombre de classes du corps K(μp). Sous l'hypothèse que p est K-régulier et inerte dans K, on établit le second cas du théorème de Fermat sur K pour l'exposant p. On utilise pour cela des arguments classiques, ainsi que le théorème de Faltings selon lequel une courbe de genre au moins deux sur K n'a qu'un nombre fini de points K-rationnels. De plus, si K est un corps quadratique imaginaire, distinct de [Formula: see text], on en déduit un énoncé permettant souvent en pratique de démontrer le théorème de Fermat sur K pour un exposant K-régulier donné. Mots-clés: Théorème de Fermat; corps de nombres. Let K be a number field and p a prime number ≥5. Let us denote by μp the group of the pth roots of unity. We define p to be K-regular if p does not divide the class number of the field K(μp). Under the assumption that p is K-regular and inert in K, we establish the second case of Fermat's Last Theorem over K for the exponent p. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over K has a finite number of K-rational points. Moreover, if K is an imaginary quadratic field, other than [Formula: see text], we deduce a statement which allows often in practice to prove Fermat's Last Theorem over K for a given K-regular exponent.

Explicit Classifications of Some 2-Extensions of a Field of Characteristic Different from 2

1990 ◽  
Vol 42 (5) ◽  
pp. 825-855 ◽  
Author(s):  
Ian Kiming
Keyword(s):  
Finite Group ◽  
Galois Group ◽  
Prime Number ◽  
Roots Of Unity ◽  
Normal Extension ◽  
Image Position ◽  
A Finite Group

Let p be a prime number. Let k be a field of characteristic different from p and containing the p-th roots of unity. Let be a finite group. Let L/k be a finite normal extension with Galois group and let c be a 2-cocycle on with coefficients in , where acts trivially on By Emb(L/k, c) we denote the question of the existence of a finite normal extension M of k, such that M contains L, such that [M: L] = p, and such that, denoting by the Galois group of M/k, the extension is given by the class of c.

scholarly journals On Block Idempotents of Modular Group Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 429-433 ◽  
Author(s):  
Masaru Osima
Keyword(s):  
Prime Number ◽  
Finite Order ◽  
Number Field ◽  
Algebraic Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Roots Of Unity ◽  
Group Rings ◽  
Irreducible Characters

We consider a group G of finite order g = pag′ where p is a prime number and (p, g′) = 1. Let Ω be the algebraic number field which contains the p-th roots of unity. Let K1, K2,…, Kn be the classes of conjugate elements in G and the first m(≦n) classes be p-regular. There exist n distinct (absolutely) irreducible characters x1, x2,…, xn of G.

scholarly journals Brauer points on Fermat curves

2001 ◽  
Vol 63 (3) ◽  
pp. 393-406
Author(s):  
William G. McCallum
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Rational Points ◽  
Roots Of Unity ◽  
Brauer Group ◽  
90Th Birthday ◽  
Fermat Curve ◽  
Fermat Curves ◽  
Natural Way

In honour of George Szekeres on his 90th birthdayIf X is a variety over a number field K, the set of K-rational points on X is contained in the subset of the adelic points cut out by the Brauer group; we call this set the set of Brauer points on the variety. If S is a set of valuations of K, we also define S-Brauer points in a natural way. It is natural to ask how good a bound on the rational points is provided by the Brauer (or S-Brauer) points.Let p > 3 be a prime number, and let X be the Fermat curve of degree p, xp + yp = 1. Let K be the field of p-th roots of unity, and let r be the p-rank of the class group of K. In this paper we show that if r < (p + 3)/8, then the set of p-Brauer points on X has cardinality at most p. We construct elements of the Brauer group of X by relating it to the Weil-Chatelet group of the jacobian of X, then use the method of Coleman and Chabauty to bound the points cut out by these elements.

Division Algebras of Prime Degree and Maximal Galois p-Extensions

2007 ◽  
Vol 59 (3) ◽  
pp. 658-672
Author(s):  
J. Mináč ◽  
A. Wadsworth
Keyword(s):  
Prime Number ◽  
Division Algebra ◽  
Roots Of Unity ◽  
Division Algebras ◽  
Prime Degree ◽  
Cyclic Algebra ◽  
Residue Fields ◽  
Residue Characteristic

AbstractLet p be an odd prime number, and let F be a field of characteristic not p and not containing the group μp of p-th roots of unity. We consider cyclic p-algebras over F by descent from L = F(μp). We generalize a theorem of Albert by showing that if μpn ⊆ L, then a division algebra D of degree pn over F is a cyclic algebra if and only if there is d ∈ D with dpn ∈ F – Fp. Let F(p) be the maximal p-extension of F. We show that F(p) has a noncyclic algebra of degree p if and only if a certain eigencomponent of the p-torsion of Br(F(p)(μp)) is nontrivial. To get a better understanding of F(p), we consider the valuations on F(p) with residue characteristic not p, and determine what residue fields and value groups can occur. Our results support the conjecture that the p torsion in Br(F(p)) is always trivial.

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