Definition talk:Galois Group of Field Extension

Galois group defined for non-Galois extensions
Exactly the same as Definition:Automorphism Group of Field Extension. They have to be merged. The central issue: Some authors only refer to the "Automorphism Group" ($\operatorname{Aut}$) as the "Galois group" ($\operatorname{Gal}$) if the extension is Galois. (Same story with Galois covering maps.) I don't know a good reason why, sometimes the authors explain that this distinction is to emphasize when an extension is Galois. (Dummit&Foote, $\S$14.1, under Proposition 5)

has to make a choice here. As guidance, a list:

Distinction: No distinction: Only consider Galois extensions/splitting fields: Don't give a name:
 * Lang - Algebra
 * Dummit, Foote - Abstract Algebra
 * Postnikov - Foundations of Galois Theory
 * Völklein - Groups as Galois Groups: An Introduction
 * van der Waerden - (Modern) Algebra (Volume I)
 * Kaplansky - Fields and rings
 * Rotman - Advanced Modern Algebra
 * Roman - Field Theory
 * Edwards - Galois Theory
 * Herstein - Topics in Algebra
 * Bewersdorff - Galois Theory for Beginners: A Historical Perspective
 * Artin - Galois Theory

Van der Waerden does something strange in the separable non-normal case, by defining the Galois group as the automorphism group of the normal closure. He then only works with Galois extensions though. (Chapter VII, $\S$50: From now on we assume that $\Sigma=K(\vartheta)$ is a normal field.) --barto (talk) 17:44, 13 July 2017 (EDT)


 * Lordy, I don't know where to start. I've only got one of those, it's the Artin work. I don't even recognise the names. I've got to improve my library.
 * Let me go away and accumulate some sources, trouble is it will take a while, I don't have a whole lot of money so I will need to save up. --prime mover (talk) 18:00, 13 July 2017 (EDT)
 * I redirected Definition:Automorphism Group of Field Extension to here and added a line about the distinction. --barto (talk) (contribs) 16:31, 15 January 2018 (EST)