scholarly journals INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS

2008 ◽  
Vol 50 (1) ◽  
pp. 27-32 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Cyclotomic Field ◽  
Galois Groups ◽  
Cyclotomic Fields ◽  
Class Field ◽  
Class Field Towers ◽  
Almost All ◽  
Divisibility Properties ◽  
Positive Integers ◽  
Hilbert Class Field ◽  
Class Field Tower

AbstractWe use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field $\Q(\exp(2 \pi i/m))$ has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over $\Q(\exp(2 \pi i/m))$ for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of $\Q(\exp(2 \pi i/m))$.

DISTRIBUTION OF GALOIS GROUPS OF MAXIMAL UNRAMIFIED 2-EXTENSIONS OVER IMAGINARY QUADRATIC FIELDS

2018 ◽  
Vol 237 ◽  
pp. 166-187
Author(s):  
SOSUKE SASAKI
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Class Field ◽  
Class Field Towers ◽  
Generalized Quaternion ◽  
Class Field Tower

Let $k$ be an imaginary quadratic field with $\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert $2$-class field tower is at least $2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J. 40(1) (1998), 63–69) calculated the density of $k$ where the length of the tower is $1$; that is, the maximal unramified $2$-extension is a $V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

scholarly journals Class numbers of real cyclotomic fields of composite conductor

2014 ◽  
Vol 17 (A) ◽  
pp. 404-417 ◽  
Author(s):  
John C. Miller
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Upper Bounds ◽  
Class Numbers ◽  
Prime Ideals ◽  
Cyclotomic Fields ◽  
Class Field ◽  
Divisibility Properties ◽  
Composite Conductor ◽  
Hilbert Class Field

AbstractUntil recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

scholarly journals INFINITE HILBERT CLASS FIELD TOWERS FROM GALOIS REPRESENTATIONS

2011 ◽  
Vol 07 (01) ◽  
pp. 1-8
Author(s):  
KIRTI JOSHI ◽  
CAMERON MCLEMAN
Keyword(s):  
Galois Representations ◽  
Number Fields ◽  
Modular Representations ◽  
Infinite Class ◽  
Fixed Field ◽  
Class Field ◽  
Class Field Towers ◽  
K 12 ◽  
Hilbert Class Field ◽  
Class Field Tower

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each k ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes ℓ such that the fixed field of the mod-ℓ representation attached to the unique normalized cusp eigenform of weight k on SL2(ℤ) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes ℓ for each k in the list. Finally, given a non-CM curve E/ℚ, we show that there exists an integer ME such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to ME.

On Cyclic Fields of Odd Prime Degree p with Infinite Hilbert p-Class Field Towers

2002 ◽  
Vol 45 (1) ◽  
pp. 86-88 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Rational Numbers ◽  
Cyclic Extension ◽  
Prime Degree ◽  
Class Field ◽  
Positive Proportion ◽  
Class Field Towers ◽  
Class Field Tower

AbstractLet k be a cyclic extension of odd prime degree p of the field of rational numbers. If t denotes the number of primes that ramify in k, it is known that the Hilbert p-class field tower of k is infinite if t > 3 + 2 . For each t > 2 + , this paper shows that a positive proportion of such fields k have infinite Hilbert p-class field towers.

On Hilbert class field tower for some quartic number fields

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohamed Talbi ◽  
Mohammed Talbi
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Class Field ◽  
Hilbert Class Field ◽  
Class Field Tower

We determine the Hilbert 2-class field tower for some quartic number fields k whose 2-class group Ck,2 is isomorphic to ℤ/2ℤ×ℤ/2ℤ.

scholarly journals Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields

1993 ◽  
Vol 48 (3) ◽  
pp. 379-383 ◽  
Author(s):  
Elliot Benjamin
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Class Field ◽  
Imaginary Quadratic Number ◽  
Hilbert Class Field ◽  
Class Field Tower ◽  
The Ideal

Letkbe an imaginary quadratic number field and letk1be the 2-Hilbert class field ofk. IfCk,2, the 2-Sylow subgroup of the ideal class group ofk, is elementary and |Ck,2|≥ 8, we show thatCk1,2is not cyclic. IfCk,2is isomorphic toZ/2Z×Z/4ZandCk1,2is elementary we show thatkhas finite 2-class field tower of length at most 2.

scholarly journals The Proof of a Conjecture Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2638
Author(s):  
Eva Trojovská ◽  
Kandasamy Venkatachalam
Keyword(s):  
Positive Integer ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Divisibility Properties ◽  
Positive Integers

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

Asymptotic Results for Class Number Divisibility in Cyclotomic Fields

1983 ◽  
Vol 26 (4) ◽  
pp. 464-472 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Lower Bounds ◽  
Class Number ◽  
Cyclotomic Field ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Asymptotic Results ◽  
Root Of Unity ◽  
Almost All

AbstractLet n ≥3 and m≥3 be integers. Let Kn be the cyclotomic field obtained by adjoining a primitive nth root of unity to the field of rational numbers. Let denote the maximal real subfield of Kn. Let hn (resp., ) denote the class number of Kn (resp., ). For fixed m we show that m divides hn and hn for asymptotically almost all n. Also for those Kn and with a given number of ramified primes, we obtain lower bounds for certain types of densities for m dividing hn and .

scholarly journals Infinite class field towers

2009 ◽  
Vol 344 (4) ◽  
pp. 923-928 ◽  
Author(s):  
Jing Long Hoelscher
Keyword(s):  
Infinite Class ◽  
Class Field ◽  
Class Field Towers
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