The Absolute Galois Group of a Semi-local Field

📘 Book Details

• Author: Dan Haran

• Publisher: Springer

• Language: English

• Edition: 2021 Edition

• ISBN: 9783030891930

• Number of Pages: 137

• Cover: Paperback

• Dimensions: 9.2 x 6.1 x 0.5 inches

📝 About the Book

The Structure of Absolute Galois Groups of Certain Algebraic Extensions by Dan Haran is a highly specialized book focused on the structure of the absolute Galois groups of algebraic extensions of the rational numbers. Published by Springer on November 21, 2022, this 137-page paperback is an essential read for researchers in the fields of Galois theory, field arithmetic, and profinite groups.

The book’s main result, a theorem co-authored by Haran and Florian Pop in 2012, provides a detailed description of the absolute Galois group of distinguished semi-local algebraic extensions of the rational numbers. It characterizes this group as a free product of the free profinite group on countably many generators and local Galois groups, answering the generalized inverse problem of Galois theory in a positive way.

The book adopts both an arithmetic and probabilistic approach, meticulously setting out the preliminary material necessary to prove the main theorem and its supporting results. Additionally, it introduces Melnikov's construction of free products of profinite groups and presents, for the first time in book form, a generalization of the theory of free products of profinite groups and their subgroups.

This work is a valuable resource for researchers and academics in field arithmetic, Galois theory, and profinite groups who seek an in-depth understanding of the structure of Galois groups and the latest advancements in the field.