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arXiv · 1909.03407

5-Class towers of cyclic quartic fields arising from quintic reflection

Abdelmalek Azizi·Yasuhiro Kishi·Daniel C. Mayer·Mohamed Talbi·Mohammed Talbi
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Abstract

Let zeta5 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((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The precise length ell5(M) of the 5-class tower of M is determined, when G(5,2)M is of order less than or equal to 5^5. Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0<d<10^4, respectively -2*10^5<d<0.

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Abdelmalek Azizi, Yasuhiro Kishi, Daniel C. Mayer, Mohamed Talbi, Mohammed Talbi. 2019-09-08. 5-Class towers of cyclic quartic fields arising from quintic reflection. https://arxiv.org/abs/1909.03407

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