scholarly journals Annihilators of the ideal class group of a cyclic extension of global function field

2021 ◽  
Author(s):  
Pascal Stucky
Keyword(s):  
Function Field ◽  
Class Group ◽  
Cyclic Extension ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Global Function ◽  
Global Function Field ◽  
The Ideal

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 $L$.

On l-class groups of global function fields

2016 ◽  
Vol 12 (02) ◽  
pp. 341-356
Author(s):  
Zhengjun Zhao ◽  
Wanbao Hu
Keyword(s):  
Class Group ◽  
Function Fields ◽  
Integral Closure ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Prime Degree ◽  
Global Function ◽  
Global Function Fields ◽  
Genus Theory ◽  
The Ideal

Let [Formula: see text] be a finite geometric separable extension of the rational function field [Formula: see text], and let [Formula: see text] be a finite cyclic extension of [Formula: see text] of prime degree [Formula: see text]. Assume that the ideal class number of the integral closure [Formula: see text] of [Formula: see text] in [Formula: see text] is not divisible by [Formula: see text]. Using genus theory and Conner–Hurrelbrink exact hexagon for function fields, we study in this paper the [Formula: see text]-class group of [Formula: see text] (i.e. the Sylow [Formula: see text]-subgroup of the ideal class group of [Formula: see text]) as Galois module, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. The resulting conclusion is used to discuss the relations of class numbers for the biquadratic function fields with their quadratic subfields.

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

2005 ◽  
Vol 48 (4) ◽  
pp. 576-579 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Natural Homomorphism ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Integral Basis ◽  
Rings Of Integers ◽  
Integer Ring ◽  
The Ideal

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

Circulant Graphs and 4-Ranks of Ideal Class Groups

1994 ◽  
Vol 46 (1) ◽  
pp. 169-183 ◽  
Author(s):  
Jurgen Hurrelbrink
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Number Fields ◽  
Regular Graphs ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Ideal Class Groups ◽  
Yield Information ◽  
The Ideal

AbstractThis is about results on certain regular graphs that yield information about the structure of the ideal class group of quadratic number fields associated with these graphs. Some of the results can be formulated in terms of the quadratic forms x2 + 27y2, x2 + 32y2, x2 + 64y2.

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.

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