The finitude of tamely ramified pro-p extensions of number fields with cyclic p-class groups

2024/02/13 by Lee, Yoonjin, Lim, Donghyeok
#11R32 #11R37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2402.08512

Abstract

Let p be an odd prime and F be a number field whose p-class group is cyclic. Let F_\\mathfrakq\ be the maximal pro-p extension of F which is unramified outside a single non-p-adic prime ideal \mathfrakq of F. In this work, we study the finitude of the Galois group G_\\mathfrakq\(F) of F_\\mathfrakq\ over F. We prove that G_\\mathfrakq\(F) is finite for the majority of \mathfrakq's such that the generator rank of G_\\mathfrakq\(F) is two, provided that for p = 3, F is not a complex quartic field containing the primitive third roots of unity.

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