Springer Algebra and Galois Theories

Galois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, starting with that of coverings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of Galois theory the geometric intuition that one can have in the context of coverings. This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory. Review “This book covers a lot of interesting material and is surely a valuable addition to the literature, but is certainly not for the timid. It brings together a broad array of sophisticated mathematics … and it does so in a very general and abstract way, with an exposition that gives whole new meaning to the word ‘concise’.” (Mark Hunacek, MAA Reviews, April 5, 2021) From the Back Cover Galois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, starting with that of coverings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of Galois theory the geometric intuition that one can have in the context of coverings.This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory. About the Author Adrien DOUADY was born in 1935. After receiving his Ph.d. in mathematics in the field of complex analytic geometry, he later joined the University Paris-Sud (Orsay). A recipient of the Ampere prize, he was a member of the French Academy of Sciences, as well as of the highly influential informal Bourbaki group. Throughout his life, he remained interested in several areas. Yet his odyssey always brought back to complex numbers. He passed away in 2006, and is notably survived by his wife, Regine Douady.  Regine Douady, born in 1934,  after a Ph.d. in the didactics of mathematics, she became a lecturer at the University Paris Denis-Diderot, as well as the head of the IREM (Institut de Recherche sur l' Enseignement des Mathematiques). She was made a chevalier dans l'ordre des palmes academiques. Now retired, she has ceaselessly endeavoured to introduce into teaching the necessity to address concepts from different standpoints-the motivating idea behind this book.