On the rank of the 2-class group of ℚ(√(p), √(q),√(-1))

2014/03/04 by Abdelmalek Azizi Mohammed Taous, Taous, Abdelmalek Azizi Mohammed, Abdelkader Zekhnini +1
Computer Science · Mathematics · #11R11 #11R29 #11R32 #11R37 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R11 #msc:11R29 #msc:11R32 #msc:11R37

paper · pdf · doi:10.48550/arxiv.1403.0662

arxiv created 2014/03/04 · openalex publication_date 2014/03/04 · arxiv updated 2014/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let d be a square-free integer, k=ℚ(√ d, i) and i=√(-1). Let k₁⁽²⁾ be the Hilbert 2-class field of k, k₂⁽²⁾ be the Hilbert 2-class field of k₁⁽²⁾ and G=Gal(k₂⁽²⁾/k) be the Galois group of k₂⁽²⁾/k. Our goal is to give necessary and sufficient conditions to have G metacyclic in the case where d=pq, with p and q are primes such that p≡ 1\pmod 8 and q≡ 5\pmod 8 or p≡ 1\pmod 8 and q≡ 3\pmod 4.

Related

• Capitulation des 2-classes d'idéaux de type (2, 4)
• On the unit group and the 2-class number of ℚ(√(2),√(p),√(q))
• Principalization of 2-class groups of type (2,2,2) of biquadratic fields ℚ(√(\strut p₁p₂q),√(\strut -1))
• On the 2-rank and 4-rank of the class group of some real pure quartic number fields
• On the unit group of some multiquadratic fields