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2024 | OriginalPaper | Chapter

Note on the Rank of 2-Class Group of \(\mathbb {Q}(i,\sqrt{p}, \sqrt{d})\)

Authors : Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini

Published in: Class Groups of Number Fields and Related Topics

Publisher: Springer Nature Singapore

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Abstract

The chapter delves into the 2-class group of number fields, particularly focusing on the 2-rank of the class group of fields defined by a prime number and a square-free integer. It builds on the ambiguous class number formula and previous research on quartic and triquadratic number fields. The main theorem provides explicit conditions and formulas for the 2-rank of the class group, differentiating between cases where the integer is prime or composite. The proof involves key lemmas that explore the ramification of prime ideals and the behavior of norm residue symbols. The chapter concludes with numerical examples obtained using Pari/gp, showcasing the practical application of the theoretical findings.
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previous chapter Congruent Numbers and Class Groups of Associated Quadratic Fields
next chapter Parameterized Families of Quadratic Fields with n-Rank at Least 2
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Metadata
Title
Note on the Rank of 2-Class Group of
Authors
Mohamed Mahmoud Chems-Eddin
Abdelkader Zekhnini
Copyright Year
2024
Publisher
Springer Nature Singapore
Book
Class Groups of Number Fields and Related Topics

Print ISBN: 978-981-9769-10-0

Electronic ISBN: 978-981-9769-11-7

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-981-97-6911-7_8

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