Explicit Constructions of the non-Abelian p³-Extensions Over \QQ

2008/12/11 by Oz Ben-Shimol, Ben-Shimol, Oz
Computer Science · Engineering · Mathematics · #11R18 #12F12 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #graph theory and CDMA systems #math.NT #msc:11R18 #msc:12F12

paper · pdf · doi:10.48550/arxiv.0812.2167

12 pages. keywords: Constructive Galois Theory, Heisenberg group, Explicit Embedding problem, Minimal Ramification

arxiv created 2008/12/11 · openalex publication_date 2008/12/11 · arxiv updated 2012/03/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime. Let F/k be a cyclic extension of degree p and of characteristic different from p. The explicit constructions of the non-abelian p³-extensions over k, are induced by certain elements in F(μ_p)^*. In this paper we let k=\QQ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over \QQ are constructed. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions.

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• Explicit Constructions of the non-Abelian p³-Extensions Over \QQ
• Non-Abelian extensions of degree p³ and p⁴ in characteristic p>2
• The geometry of the moduli Space of non-cyclic biquadratic field extensions
• Upper ramification sequences of nonabelian extensions of degree p³ in characteristic p
• Abelian p-extensions and additive polynomials