Tame kernels and second regulators of number fields and their subfields

2013 ◽  
Vol 12 (1) ◽  
pp. 137-165 ◽  
Author(s):  
Jerzy Browkin ◽  
Herbert Gangl
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Number Field ◽  
Dihedral Group ◽  
Number Fields ◽  
Numerical Results ◽  
Alternating Group ◽  
Dedekind Zeta Function ◽  
Tame Kernel ◽  
Tame Kernels

AbstractAssuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number field F with the orders of the tame kernels of some of its subfields. The details are given for fields F which are Galois over ℚ with Galois group the group ℤ/2 × ℤ/2, the dihedral group D2p; p an odd prime, or the alternating group A4. We include numerical results illustrating these formulas.

scholarly journals ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS

2015 ◽  
Vol 93 (2) ◽  
pp. 199-210 ◽  
Author(s):  
ANDREA FERRAGUTI ◽  
GIACOMO MICHELI
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Dedekind Zeta Function ◽  
Natural Density ◽  
Suitable Notion

Let $K$ be a number field with ring of integers ${\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\mathcal{O}}$, generalising the natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\it\zeta}_{K}(m)$, where ${\it\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\mathbb{Z}$.

A note on values of the Dedekind zeta-function at odd positive integers

2021 ◽  
pp. 1-12
Author(s):  
M. Ram Murty ◽  
Siddhi S. Pathak
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Imaginary Quadratic Fields ◽  
Fixed Integer ◽  
Dedekind Zeta Function ◽  
Totally Real ◽  
Positive Integers ◽  
Real Number Field

For an algebraic number field [Formula: see text], let [Formula: see text] be the associated Dedekind zeta-function. It is conjectured that [Formula: see text] is transcendental for any positive integer [Formula: see text]. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when [Formula: see text] is a totally real number field, [Formula: see text] is an algebraic multiple of [Formula: see text] and hence, is transcendental. If [Formula: see text] is not totally real, the question of whether [Formula: see text] is irrational or not remains open. In this paper, we prove that for a fixed integer [Formula: see text], at most one of [Formula: see text] is rational, as [Formula: see text] varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields.

scholarly journals Class-number problems for cubic number fields

1995 ◽  
Vol 138 ◽  
pp. 199-208 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Class Number ◽  
Number Fields ◽  
Algebraic Integers ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
Cubic Number Fields ◽  
Ring Of Algebraic Integers ◽  
The Absolute

Let M be any number field. We let DM, dM, hu, , AM and RegM be the discriminant, the absolute value of the discriminant, the class-number, the Dedekind zeta-function, the ring of algebraic integers and the regulator of M, respectively.we set If q is any odd prime we let (⋅/q) denote the Legendre’s symbol.

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

ON FINITE GALOIS STABLE SUBGROUPS OF GLn IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS

2009 ◽  
Vol 08 (04) ◽  
pp. 493-503 ◽  
Author(s):  
H.-J. BARTELS ◽  
D. A. MALININ
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Galois Extension ◽  
Commutator Subgroup ◽  
Finite Extension ◽  
Number Fields ◽  
Maximal Order ◽  
Linear Groups ◽  
Stable Linear

Let K/ℚ be a finite Galois extension with maximal order [Formula: see text] and Galois group Γ. For finite Γ-stable subgroups [Formula: see text] it is known [4], that they are generated by matrices with coefficients in [Formula: see text], Kab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups [Formula: see text] which are not fixed elementwise by the commutator subgroup of Gal (K/R).

scholarly journals EXPLICIT ZERO-FREE REGIONS FOR DEDEKIND ZETA FUNCTIONS

2012 ◽  
Vol 08 (01) ◽  
pp. 125-147 ◽  
Author(s):  
HABIBA KADIRI
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Zeta Functions ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
The Absolute

Let K be a number field, nK be its degree, and dK be the absolute value of its discriminant. We prove that, if dK is sufficiently large, then the Dedekind zeta function ζK(s) has no zeros in the region: [Formula: see text], [Formula: see text], where log M = 12.55 log dK + 9.69nK log |ℑ𝔪 s| + 3.03 nK + 58.63. Moreover, it has at most one zero in the region:[Formula: see text], [Formula: see text]. This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: ζK(s) has at most one zero in the region [Formula: see text], [Formula: see text].

scholarly journals New number fields with known p-class tower

2015 ◽  
Vol 64 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Finite Index ◽  
Number Fields ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Abstract The p-class tower F∞p (k) of a number field k is its maximal unramified pro-p extension. It is considered to be known when the p-tower group, that is the Galois group G := Gal(F∞p (k)|k), can be identified by an explicit presentation. The main intention of this article is to characterize assigned finite 3-groups uniquely by abelian quotient invariants of subgroups of finite index, and to provide evidence of actual realizations of these groups by 3-tower groups G of real quadratic fields K = ℚ( √d) with 3-capitulation type (0122) or (2034).

Galois groups of doubly even octic polynomials

2019 ◽  
Vol 19 (01) ◽  
pp. 2050014
Author(s):  
Anna Altmann ◽  
Chad Awtrey ◽  
Sam Cryan ◽  
Kiley Shannon ◽  
Madeleine Touchette
Keyword(s):  
Conjugacy Class ◽  
Galois Group ◽  
Number Field ◽  
Cyclic Group ◽  
Dihedral Group ◽  
Irreducible Polynomial ◽  
Galois Groups ◽  
General Knowledge ◽  
Split Extension

Let [Formula: see text] be an irreducible polynomial with rational coefficients, [Formula: see text] the number field defined by [Formula: see text], and [Formula: see text] the Galois group of [Formula: see text]. Let [Formula: see text], and let [Formula: see text] be the Galois group of [Formula: see text]. We investigate the extent to which knowledge of the conjugacy class of [Formula: see text] in [Formula: see text] determines the conjugacy class of [Formula: see text] in [Formula: see text]. We show that, in general, knowledge of [Formula: see text] does not automatically determine [Formula: see text], except when [Formula: see text] is isomorphic to [Formula: see text] (the cyclic group of order 4). In this case, we show [Formula: see text] is isomorphic to a non-split extension of [Formula: see text] (the dihedral group of order 8) by [Formula: see text]. We also show that [Formula: see text] is completely determined when [Formula: see text] is isomorphic to [Formula: see text] and [Formula: see text] is a perfect square. In this case, [Formula: see text].

A note on Dedekind zeta values at 1/2

2021 ◽  
pp. 1-11
Author(s):  
Neelam Kandhil
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Number Field ◽  
Dedekind Zeta Function ◽  
Derivative Formula ◽  
Zeta Values

For a number field [Formula: see text], let [Formula: see text] be the Dedekind zeta function associated to [Formula: see text]. In this paper, we study non-vanishing and transcendence of [Formula: see text] as well as its derivative [Formula: see text] at [Formula: see text]. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of [Formula: see text] and non-vanishing of [Formula: see text]-series at [Formula: see text], J. Number Theory 161 (2016) 444–456].

scholarly journals On -invariants attached to cyclic cubic number fields

2015 ◽  
Vol 18 (1) ◽  
pp. 684-698
Author(s):  
Daniel Delbourgo ◽  
Qin Chao
Keyword(s):  
Power Series ◽  
Zeta Function ◽  
Galois Group ◽  
Prime Number ◽  
Class Number ◽  
Class Group ◽  
Number Fields ◽  
Cubic Number Fields

We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$, where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$-invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$. If the ${\it\lambda}$-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$-adic $L$-function and deduce ${\rm\Lambda}$-monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.Supplementary materials are available with this article.

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