On the capitulation problem of some pure metacyclic fields of degree 20

2020/10/29 by Elmouhib, Fouad, Talbi, Mohamed, Azizi, Abdelmalek
#11R04 #11R18 #11R29 #11R37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2010.15935

Abstract

Let Γ = ℚ(√[5]n) be a pure quintic field, where n is a positive integer 5^th power-free, k₀ = ℚ(ζ₅) be the cyclotomic field containing a primitive 5^th root of unity ζ₅, and k = ℚ(√[5]n,ζ₅) the normal closure of Γ. Let k₅⁽¹⁾ be the Hilbert 5-class field of k, C_k,5 the 5-ideal classes group of k, and C_k,5^(σ) the group of ambiguous classes under the action of Gal(k/k₀) = ⟨σ⟩. When C_k,5 is of type (5,5) and rank C_k,5^(σ) = 1, we study the capitulation problem of the 5-ideal classes of C_k,5 in the six intermediate extensions of k₅⁽¹⁾/k.

Related

• On the capitulation problem of some pure metacyclic fields of degree 20 II
• The generators of 5-class group of some fields of degree 20 over ℚ
• On the exact divisibility by 5 of the class number of some pure metacyclic fields
• Capitulation in the absolutely abelian extensions of some number fields II
• Capitulation in the absolutely abelian extensions of some fields ℚ(√(p₁p₂q), √(-1))