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

Yoonjin Lee, Donghyeok Lim

Abstract

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

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

Abstract

Let be an odd prime and be a number field whose -class group is cyclic. Let be the maximal pro- extension of which is unramified outside a single non--adic prime ideal of . In this work, we study the finitude of the Galois group of over . We prove that is finite for the majority of 's such that the generator rank of is two, provided that for , is not a complex quartic field containing the primitive third roots of unity.
Paper Structure (8 sections, 21 theorems, 35 equations)

This paper contains 8 sections, 21 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. The finitude of powerful tame pro-$p$ groups
  3. Powerfulness of $G_{\{\mathfrak{q}\}}(F)$
  4. An application of the Gras-Munnier theorem to the case when $\mathfrak{q}$ splits in $F_{1}$
  5. Gras-Munnier theorem
  6. Proof of Theorem \ref{['maintheorem2']}
  7. The $\mathbb{F}_{p}[\mathrm{Gal}(F_{1}/F)]$-module structure of $\mathrm{Gal}(\mathrm{Gov}(F_{1})/F_{1}(\zeta_{p}))$
  8. $\mathbb{F}_{p}[\mathrm{Gal}(F_{1}/F)]$-module structure of $V_{\emptyset}(F_{1})/F_{1}^{\times \, p}$

Key Result

Theorem 1.1

Let $F$ be a number field whose $p$-class group is non-trivial and cyclic. Let $\mathfrak{q}$ be a non-$p$-adic prime of $F$ which does not split in the $p$-class field tower of $F$. Then $G_{\{\mathfrak{q}\}}(F)$ is powerful and finite.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 36 more