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An infinite family of pure quartic fields with class number $\equiv 2\pmod{4}$
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In this paper, the authors consider pure quartic fields of the form (K=\Q(\sqrt[4]{p})$ where 0 <p\equiv 7\pmod{16}$ is a prime integer.Abstract:
Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$, then the Hilbert class field of $\K$ is $\H_\K=\K(\sqrt{2})$. Finally, we find a criterion to decide if an ideal of the ring of integers or $\K$ is principal or non-principal.read more
Citations
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Déterminations des corps $K = mathbb Q(\surdd, \surd −1)$ dont les 2-groupes de classes sont de type (2, 4) ou (2, 2, 2)
Abdelmalek Azizi,Mohammed Taous +1 more
Les corps $\mathbb{Q}\,\left( \sqrt{-p_{o}},\sqrt{d} \right)$ dont les 2-groupes de classes sont de Klein, avec $p_{o}\, \equiv \, 1\, \mbox{ mod(4)}$, premier
A. Azizi,R. Lamjoun +1 more
Journal ArticleDOI
On efficient computation of the 2-parts of ideal class groups of quadratic fields
Julius M. Basilla,Hideo Wada +1 more
TL;DR: In this article, a relation between Gauss' ternary quadratic form and an ideal of a Quadratic Field has been shown, which can be used to compute the 2-part ideal class group.
References
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Book
Algebraic Number Theory
TL;DR: In this paper, Algebraic integral integers, Riemann-Roch theory, Abstract Class Field Theory, Local Class Field theory, Global Class Field and Zeta Functions are discussed.
Book
Algebraic Number Theory
TL;DR: This edition of Algebraic Number Theory, Second Edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter.
Book
Introductory Algebraic Number Theory
Şaban Alaca,Kenneth S. Williams +1 more
TL;DR: Algebraic number theory is a subject which came into being through the attempts of mathematicians to try to prove Fermat's last theorem and which now has a wealth of applications to diophantine equations, cryptography, factoring, primality testing and public-key cryptosystems as mentioned in this paper.
Journal ArticleDOI
Imaginary Bicyclic Biquadratic Fields With Cyclic 2-Class Group
TL;DR: In this paper, a method for determining the rank of the 2-class group of imaginary bicyclic biquadratic fields is described, which is used to determine all such fields with cyclic 2class group.
Journal Article
The 2-class group of certain biquadratic number fields.
Ezra Brown,Charles J. Parry +1 more
TL;DR: In this paper, the exact power of 2 dividing the class number of a cyclic biquadratic field has been shown to be polynomial in the number of vertices.
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