scholarly journals On the divisibility of the class number of by 16

1983 ◽  
Vol 26 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Philip A. Leonard ◽  
Kenneth S. Williams
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Sufficient Conditions ◽  
Ideal Class Group ◽  
Squarefree Integer ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
The Ideal

Let d(<0) denote a squarefree integer. The ideal class group of the imaginary quadratic field has a cyclic 2-Sylow subgroup of order ≦8 in precisely the following cases (see for example [5] and [6]):where p and q denote primes and g, h, u and v are positive integers. The class number of is denoted by h(d) and in the above cases h(d) = 0(mod 8). For cases (i), (ii) and (iii) the authors [6] have given necessary and sufficient conditions for h(d) to be divisible by 16. In this paper we do the same for case (iv) extending the results of Brown [4].

scholarly journals ON THE IDEAL CLASS GROUP OF CERTAIN QUADRATIC FIELDS

2010 ◽  
Vol 52 (3) ◽  
pp. 575-581 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Ideal Class Group ◽  
The Real ◽  
Real Quadratic ◽  
The Ideal

AbstractLet n(≥ 3) be an odd integer. Let k:= $\Q(\sqrt{4-3^n})\)$ be the imaginary quadratic field and k′:= $\Q(\sqrt{-3(4-3^n)})\)$ the real quadratic field. In this paper, we prove that the class number of k is divisible by 3 unconditionally, and the class number of k′ is divisible by 3 if n(≥ 9) is divisible by 3. Moreover, we prove that the 3-rank of the ideal class group of k is at least 2 if n(≥ 9) is divisible by 3.

scholarly journals Note on the class-number of the maximal real subfield of a cyclotomic field, II

1989 ◽  
Vol 113 ◽  
pp. 147-151
Author(s):  
Hiroyuki Osada
Keyword(s):  
Natural Number ◽  
Class Number ◽  
Class Group ◽  
Sufficient Conditions ◽  
Cyclotomic Field ◽  
Sufficient Condition ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Root Of Unity ◽  
The Ideal

For an integer m > 2, we denote by C(m) and H(m) the ideal class group and the class-number of the fieldK = Q(ζm + ζm−1)respectively, where ζm is a primitive m-th root of unity. Let q be a prime and /Q be a real cyclic extension of degree q. Let C() and h() be the ideal class group and the class-number of . In this paper, we give a relation between C() (resp. h()) and C(m) (resp. H(m)) in the case that m is the conductor of (Main Theorem). As applications of this main theorem, we give the following three propositions. In the previous paper [4], we showed that there exist infinitely many square-free integers m satisfying n|H(m) for any given natural number n. Using the result of Nakahara [2], we give first an effective sufficient condition for an integer m to satisfy n|H(m) for any given natural number n (Proposition 1). Using the result of Nakano [3], we show next that there exist infinitely many positive square-free integers m such that the ideal class group C(m) has a subgroup which is isomorphic to (Z/nZ)2 for any given natural number n (Proposition 2). In paper [4], we gave some sufficient conditions for an integer m to satisfy 3|H(m) and m≡l (mod 4). In this paper, using the result of Uchida [5], we give moreover a sufficient condition for an integer m to satisfy 4|H(m) and m ≡ 3 (mod 4) (Proposition 3). Finally, we give numerical examples of some square-free integers m satisfying 4|H(m) and m ≡ 3 (mod 4).

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

Class Number Divisibility in Real Quadratic Function Fields

1992 ◽  
Vol 35 (3) ◽  
pp. 361-370 ◽  
Author(s):  
Christian Friesen
Keyword(s):  
Class Number ◽  
Function Field ◽  
Class Group ◽  
Quadratic Function ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Function Field ◽  
Quadratic Function Fields ◽  
Real Quadratic ◽  
The Ideal

AbstractLet q be a positive power of an odd prime p, and let Fq(t) be the function field with coefficients in the finite field of q elements. Let denote the ideal class number of the real quadratic function field obtained by adjoining the square root of an even-degree monic . The following theorem is proved: Let n ≧ 1 be an integer not divisible by p. Then there exist infinitely many monic, squarefree polynomials, such that n divides the class number, . The proof constructs an element of order n in the ideal class group.

scholarly journals Annihilators of the Ideal Class Group of a Cyclic Extension of an Imaginary Quadratic Field

2019 ◽  
Vol 71 (6) ◽  
pp. 1395-1419
Author(s):  
Hugo Chapdelaine ◽  
Radan Kučera
Keyword(s):  
Sylow Subgroup ◽  
Class Group ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Cyclic Extension ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Elliptic Units ◽  
The Ideal

AbstractThe aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.

scholarly journals Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1206 ◽  
Author(s):  
Alex Brandts ◽  
Tali Pinsky ◽  
Lior Silberman
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Finite Collection ◽  
Ideal Class Group ◽  
Modular Surface ◽  
Closed Curves ◽  
Hyperbolic Three Manifolds ◽  
Unit Tangent Bundle ◽  
Real Quadratic ◽  
The Ideal

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

On the Divisibility of the Class Numbers of Q(√−p) and Q(√−2p) by 16.

1982 ◽  
Vol 25 (2) ◽  
pp. 200-206 ◽  
Author(s):  
Philip A. Leonard ◽  
Kenneth S. Williams
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Class Numbers ◽  
Necessary And Sufficient

AbstractLet h(m) denote the class number of the quadratic field Q(√m). In this paper necessary and sufficient conditions for h (m) to be divisible by 16 are determined when m = −p, where p is a prime congruent to 1 modulo 8, and when m = −2p, where p is a prime congruent to ±1 modulo 8.

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