The Roots of Unity with Prime Number Exponent l and the Cyclotomic Field They Generate

1998 ◽  
pp. 161-165
Author(s):  
David Hilbert
Keyword(s):  
Prime Number ◽  
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.

ON THE 2-ADIC IWASAWA LAMBDA INVARIANTS OF THE p-CYCLOTOMIC FIELDS AND THEIR QUADRATIC TWISTS

2014 ◽  
Vol 10 (02) ◽  
pp. 283-296
Author(s):  
HUMIO ICHIMURA ◽  
SHOICHI NAKAJIMA ◽  
HIROKI SUMIDA-TAKAHASHI
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Relative Class ◽  
Quadratic Twists ◽  
Large N ◽  
Relative Class Number

Let p be an odd prime number, Kn = Q(ζpn+1) the pn+1th cyclotomic field and [Formula: see text] the relative class number of Kn. Fixing an integer d ∈ Z with [Formula: see text], we denote by Ln the imaginary quadratic subextension of the imaginary (2, 2)-extension [Formula: see text] with Ln ≠ Kn. When d < 0, we have [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the minus parts of the 2-adic Iwasawa lambda invariants of Kn and Ln, respectively. By a theorem of Friedman, these invariants are stable for sufficiently large n. First, under the assumption that [Formula: see text] is odd for all n ≥ 1, we give a quite explicit version of this result. Second, we show that the assumption is satisfied for all p ≤ 599. Further, using these results, we compute the invariants [Formula: see text] and [Formula: see text] with d = -1, -3 for all p ≤ 599 and all n with the help of the computer.

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.

On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field

1981 ◽  
Vol 33 (1) ◽  
pp. 55-58 ◽  
Author(s):  
Hiroshi Takeuchi
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Cyclotomic Field ◽  
Rational Numbers ◽  
Real Quadratic Field ◽  
The Real ◽  
Root Of Unity ◽  
Image Position ◽  
Real Quadratic

Let p be an integer and let H(p) be the class-number of the fieldwhere ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).We designate by h(p) the class-number of the real quadratic field

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