scholarly journals The p-adic Kummer–Leopoldt constant: Normalized p-adic regulator

2018 ◽  
Vol 14 (02) ◽  
pp. 329-337 ◽  
Author(s):  
Georges Gras
Keyword(s):  
Field Theory ◽  
Number Field ◽  
Torsion Group ◽  
Class Field Theory ◽  
Cyclotomic Fields ◽  
Class Field ◽  
Elementary Formula ◽  
Ramification Theory ◽  
Definition Of ◽  
Theory Interpretation

The [Formula: see text]-adic Kummer–Leopoldt constant [Formula: see text] of a number field [Formula: see text] is (assuming the Leopoldt conjecture) the least integer [Formula: see text] such that for all [Formula: see text], any global unit of [Formula: see text], which is locally a [Formula: see text]th power at the [Formula: see text]-places, is necessarily the [Formula: see text]th power of a global unit of [Formula: see text]. This constant has been computed by Assim and Nguyen Quang Do using Iwasawa’s techniques, after intricate studies and calculations by many authors. We give an elementary [Formula: see text]-adic proof and an improvement of these results, then a class field theory interpretation of [Formula: see text]. We give some applications (including generalizations of Kummer’s lemma on regular [Formula: see text]th cyclotomic fields) and a natural definition of the normalized [Formula: see text]-adic regulator for any [Formula: see text] and any [Formula: see text]. This is done without analytical computations, using only class field theory and especially the properties of the so-called [Formula: see text]-torsion group [Formula: see text] of Abelian [Formula: see text]-ramification theory over [Formula: see text].

scholarly journals The Notion of Restricted Idèles with Application to Some Extension Fields

1966 ◽  
Vol 27 (1) ◽  
pp. 121-132
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Field Theory ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Class Field Theory ◽  
Normal Extension ◽  
Finite Degree ◽  
Ground Field ◽  
Class Field ◽  
Abelian Extensions

Let k be an algebraic number field of finite degree, K be its normal extension of degree n, and ŝ be the set of those primes of K which have degree 1. Using this set s instead of the set of all primes of K, we define an s-restricted idèle of K by the same way as ordinary idèles. It is known by Bauer that the normal extension of an algebraic number field is determined by the set of all primes of the ground field which are decomposed completely in the extension field. This suggests that if we treat abelian extensions over K which are normal over k, the class field theory is expressed by means of the ŝ-restricted idèles (theorem 2). When K = k, ŝ is the set of all primes of K, and we have the ordinary class field theory.

SUR LA CONJECTURE FAIBLE DE GREENBERG DANS LE CAS ABÉLIEN p-DÉCOMPOSÉ

2006 ◽  
Vol 02 (01) ◽  
pp. 49-64 ◽  
Author(s):  
NGUYEN QUANG DO THONG
Keyword(s):  
Field Theory ◽  
Number Field ◽  
Weak Form ◽  
Class Field Theory ◽  
Class Field ◽  
Root Of Unity ◽  
Kummer Theory ◽  
Circular Units

Let p be an odd prime. For any CM number field K containing a primitive pth root of unity, class field theory and Kummer theory put together yield the well known reflection inequality λ+ ≤ λ- between the "plus" and "minus" parts of the λ-invariant of K. Greenberg's classical conjecture predicts the vanishing of λ+. We propose a weak form of this conjecture: λ+ = λ- if and only if λ+ = λ- = 0, and we prove it when K+ is abelian, p is totally split in K+, and certain conditions on the cohomology of circular units are satisfied (e.g. in the semi-simple case).

scholarly journals Ramification theory in non-abelian local class field theory

2010 ◽  
Vol 144 (4) ◽  
pp. 373-393 ◽  
Author(s):  
Kâzim Ilhan Ikeda ◽  
Erol Serbest
Keyword(s):  
Field Theory ◽  
Class Field Theory ◽  
Class Field ◽  
Ramification Theory ◽  
Local Class Field Theory

scholarly journals Capitulation in Class Field Extensions of Type (p, p)

1980 ◽  
Vol 32 (5) ◽  
pp. 1229-1243 ◽  
Author(s):  
S. M. Chang ◽  
R. Foote
Keyword(s):  
Field Theory ◽  
Abelian Group ◽  
Number Field ◽  
Class Field Theory ◽  
Class Groups ◽  
Class Field ◽  
Field Extensions ◽  
Class Fields ◽  
Image Position ◽  
Hilbert Class Field

Let K be a number field, K(1) its Hilbert class field, that is, the maximal abelian unramified extension of K, let K(2) be the Hilbert class field of K(1), and let G = Gal(K(2)/K) (alternatively, for p a prime the first and second p class fields enjoy properties analogous to those of the respective class fields discussed in this introduction; the particulars may be found surrounding Lemma 2). Since G/G’ is the largest abelian quotient of G, G/G′ = Gal (K(1))/K) and so G’ is the abelian group Gal(K(2)K(1)); moreover, class field theory provides (Artin) maps φK, φK(1) which are isomorphisms of the class groups Ck, Ck(1) onto G/G′, G′ respectively. In the remarkable paper [1] E. Artin computed the composition VG′where e is the homomorphism induced on the class groups by extending ideals of K to ideals of K(1), and he gave a formula for computing VG′, the now familiar transfer (Verlagerung) homomorphism, in terms of the group G alone (see Lemma 1).

scholarly journals Tribonacci numbers and primes of the form p = x2 + 11y2

2019 ◽  
Vol 69 (3) ◽  
pp. 521-532
Author(s):  
Tim Evink ◽  
Paul Alexander Helminck
Keyword(s):  
Field Theory ◽  
Prime Number ◽  
Number Field ◽  
Fourier Coefficients ◽  
Class Field Theory ◽  
Class Field ◽  
Galois Closure ◽  
Tribonacci Numbers ◽  
Exponential Formula

Abstract In this paper we show that for any prime number p not equal to 11 or 19, the Tribonacci number Tp−1 is divisible by p if and only if p is of the form x2 + 11y2. We first use class field theory on the Galois closure of the number field corresponding to the polynomial x3 − x2 − x − 1 to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for Tp−1. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight 2 and level 11.

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