Mathematical  Culture and Thought

Mathematical Culture and Thought

Class Field Theory Over a Field with One Element

Document Type : Survey

Authors
Department of Mathematical Sciences‎, ‎Sharif University of Technology‎, ‎Iran
Abstract
Class field theory provides a framework to describe all finite abelian extensions of a number field in terms of the arithmetic properties of the field itself. This paper investigates this theory, with a particular focus on the problem of cyclotomy over the field with one element. Furthermore, generalizations of this theory, including the concept of arithmetic surfaces introduced by Bloch, are examined. The main objective of this paper is to present a coherent formulation of the aspects related to Artin’s conjecture and geometry over the field with one element, in order to provide a clear picture of how the Kronecker-Weber theorem is generalized in this abstract setting.
Keywords
Subjects

[۱] درفشه، محمدرضا، اثباتی برای قانون مربعی‌گاوس، فرهنگ و اندیشه ریاضی، ۴۱ (۱۴۰۱)، شماره ۲، ۶۳-۷۲.
[۲] رستگار، آرش، پیشینۀ تاریخی نظریۀ میدان‌های رده‌ای، فرهنگ و اندیشه ریاضی، ۴۲ (۱۴۰۲)، شماره ۲، ۳۱۳-
.۳۴۲
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Volume 44, Issue 2 - Serial Number 77
October 2025
Pages 219-234

  • Receive Date 11 August 2023
  • Accept Date 21 November 2023
  • Publish Date 23 September 2025