Integral Basis for quartic Kummer extensions over ℤ[ι]

2024/10/23 by S. Venkataraman, Venkataraman, S., Manisha Kulkarni +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2410.17560

openalex publication_date 2024/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K=ℚ[ι] and N=K[√[4]α], α∈ℤ[ι], alpha=fg²h³, f, g, h∈ ℤ[ι] are pairwise coprime and square free. Let O_N be the ring of integers of N. In this article we construct normalised integral basis for O_N over ℤ[ι], that is an integral basis of the form \1,(f₁(α))/(d₁),(f₂(α))/(d₂),\fracf₃(α)d₃\ where d_i ∈ ℤ[i] and f_i(X), ≤ i≤ 3 are monic polynomials of degree i over ℤ[ι]. We explicitly determine what d_i, 1≤ i≤ n-1 are in terms of f, g and h.

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