5-Class towers of cyclic quartic fields arising from quintic reflection
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5-Class towers of cyclic quartic fields arising from quintic reflection Abdelmalek Azizi1 · Yasuhiro Kishi2 · Daniel C. Mayer3 Mohammed Talbi4
· Mohamed Talbi4 ·
Received: 29 June 2019 / Accepted: 23 September 2019 © Fondation Carl-Herz and Springer Nature Switzerland AG 2019
Abstract Let ζ5 be a primitive fifth root of unity and d = 1 be a quadratic fundamental discriminant √ not divisible by 5. For the 5-dual cyclic quartic field M = Q((ζ5 − ζ5−1 ) d) of the quadratic √ √ fields k1 = Q( d) and k2 = Q( 5d) in the sense of the quintic reflection theorem, the (2) (2) possibilities for the isomophism type of the Galois group G5 M = Gal(M5 /M) of the (2) second Hilbert 5-class field M5 of M are investigated, when the 5-class group Cl5 (M) is (∞) elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M5 of M (2) coincides with M5 already. The precise length 5 M of the 5-class tower of M is determined, (2) when G5 M is of order less than or equal to 55 . Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0 < d < 104 , respectively −2 · 105 < d < 0. Keywords 5-Class field tower · 5-Principalization · Quadratic fields · 5-Dual cyclic quartic fields · Frobenius fields; finite 5-groups · Schur σ -groups Résumé Soient ζ5 une racine primitive 5-ième de l’unité et d = 1 un discriminant fondamental √ quadratique non divisible par 5. Pour le corps quartique cyclique M = Q((ζ5 −ζ5−1 ) d), le 5√ √ dual des corps quadratiques k1 = Q( d) et k2 = Q( 5d) au sens du théorème de réflexion, (2) (2) les possibilités pour le type d’isomorphisme du groupe de Galois G5 M = Gal(M5 /M) du (2) second 5-corps de classes de Hilbert M5 de M sont examinées lorsque le 5-groupe de classes (∞) Cl5 (M) est de type (5, 5). En général, la pro-5-extension maximale non ramifiée M5 de M (2) coïncide avec M5 . La longueur précise 5 M de la tour des 5-corps de classes de Hilbert de (2) M est déterminée lorsque G5 M est d’ordre inférieur ou égal à 55 . Les résultats théoriques
Daniel C. Mayer was supported by the Austrian Science Fund (FWF): P 26008-N25.
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Daniel C. Mayer [email protected] http://www.algebra.at
Extended author information available on the last page of the article
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A. Azizi et al.
sont étayés par le calcul réel de tous les 83 (resp. 93) cas dans l’intervalle 0 < d < 104 , (resp. −2 · 105 < d < 0). Mathematics Subject Classification Primary 11R37 · 11R29 · 11R11 · 11R16 · 11R20 · 11Y40; Secondary 20D15
1 Introduction The present article arose from the desire to generalize our results [1] for the second 3√ √ (2) class group Gal(k3 /k) of the bicyclic biquadratic field k = Q( −3, d), which is the √ √ compositum of 3-dual quadratic fields k1 = Q( d) and k2 = Q( −3d) in the cubic reflection theorem, to the situation of the quintic reflection theorem. The precise statement of both reflection theorems requires the concept of virtual units. Let p be a prime number and K be a number field with multiplicative group K × = K \{0}, maximal order
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