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).

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 A Note on Ideal Class Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 239-247 ◽  
Author(s):  
Kenkichi Iwasawa
Keyword(s):  
Class Group ◽  
Cyclotomic Field ◽  
Theoretical Method ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ideal Class Groups ◽  
The Ideal

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

On the orders of ideal classes in prime cyclotomic fields

1990 ◽  
Vol 108 (2) ◽  
pp. 197-201 ◽  
Author(s):  
Francisco Thaine
Keyword(s):  
Sylow Subgroup ◽  
Class Group ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
The Ideal

In this article we exhibit a method complementary to the method presented in [4], that allows us, at least in some important cases, to obtain exact expressions for the orders of ideal classes of cyclotomic fields in terms of properties of the units of the field. We consider only the particular case in which the classes belong to the p-Sylow subgroup (A)p of the ideal class group of a real p-cyclotomic field, but it appears that the results can be generalized.

On the compositum of orthogonal cyclic fields of the same odd prime degree

2020 ◽  
pp. 1-25
Author(s):  
Cornelius Greither ◽  
Radan Kučera
Keyword(s):  
Class Number ◽  
Class Group ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Prime Degree ◽  
Circular Units ◽  
The Ideal

Abstract The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

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 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].

On minus quotients of ideal class groups of cyclotomic fields

2020 ◽  
Vol 16 (09) ◽  
pp. 2013-2026
Author(s):  
Satoshi Fujii
Keyword(s):  
Abelian Group ◽  
Class Group ◽  
Finite Abelian Group ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
The Ideal

Let [Formula: see text] be the minus quotient of the ideal class group of the [Formula: see text]th cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of [Formula: see text] for some [Formula: see text]. Second, we show that, for all pairs of integers [Formula: see text] and [Formula: see text] with [Formula: see text], the kernel of the lifting map [Formula: see text] is contained in the [Formula: see text]-torsion [Formula: see text] of [Formula: see text]. Such an evaluation of the exponent is an individuality of cyclotomic fields.

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