The ideal class group of the Z 23 -extension over the rational field

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.

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Circulant Graphs and 4-Ranks of Ideal Class Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1994-005-2 ◽

1994 ◽

Vol 46 (1) ◽

pp. 169-183 ◽

Cited By ~ 1

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.

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Class Number Divisibility in Real Quadratic Function Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-048-5 ◽

1992 ◽

Vol 35 (3) ◽

pp. 361-370 ◽

Cited By ~ 9

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.

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

Canadian Journal of Mathematics ◽

10.4153/cjm-2018-035-9 ◽

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 $L$.

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A Note on Ideal Class Groups

Nagoya Mathematical Journal ◽

10.1017/s0027763000012046 ◽

1966 ◽

Vol 27 (1) ◽

pp. 239-247 ◽

Cited By ~ 9

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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