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)
$\mathsf{Pr} \infty \mathsf{fWiki}$ has to make a choice here. As guidance, a list:
Distinction:
- 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)
No distinction:
- Kaplansky - Fields and rings
- Rotman - Advanced Modern Algebra
- Roman - Field Theory
Only consider Galois extensions/splitting fields:
- Edwards - Galois Theory
- Herstein - Topics in Algebra
- Bewersdorff - Galois Theory for Beginners: A Historical Perspective
Don't give a name:
- 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)