次数 \(\geqslant5\) 的多项式方程根式不可解的证明, 不是只有 Galois theory 这一个办法. 1963 年, V. I. Arnold(June 12, 1937-June 3, 2010) 给出了一个拓扑证明. 这种新的思想, 可以用来解决另外一些经典问题. 随后, Topological Galois theory 就发展起来了. 该理论的建立者是 Askold G. Khovanskii.
Askold G. Khovanskii 于 2008 年在莫斯科出版了一本 Topological Galois theory. 该书的英译本会被 Springer 在 2015 年上半年出版.
This book provides a detailed and largely self-contained description of various classical and new results on solvability and unsolvability of equations in explicit form. In particular, a complete exposition of topological Galois theory is given. A unique feature of this book is that recent results are presented in the same elementary manner as classical Galois theory.
Contents
Introduction
Chapter 1: Liouville’s Theory
Chapter 2: Galois Theory
Chapter 3: Picard–Vessiot Theory
Chapter 4: Coverings and Galois Theory
Chapter 5: One-Dimensional Topological Galois Theory
Chapter 6: Solvability of Fuchsian Equations
Chapter 7: Multidimensional Topological Galois Theory
Appendix 1: Ruler and Compass Constructions
Appendix 2: Chebyshev Polynomials and Their Inverses
Appendix 3: Signatures of Branched Coverings
Appendix 4: On an Algebraic Version of Hilbert’s 13th Problem
Index
Olivia Caramello 在 arXiv 有一篇 82 页的 Topological Galois theory, 介绍了一种 topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts.
现在又兴起了所谓的 Semi-topological Galois theory.