An additional question: Is it correct that Chevalley only gave a proof for reflection groups (reflection = pseudo-reflection of order 2) in this paper and that Serre later realized that the proof also works for pseudo-reflection groups?

choice. So the style of definition in Quing's book (which gives the definition in a restricted situation which is probably easier) indeed makes sense as long as one mentions the extra conditions all along.

@Georges: Moreover, thanks to your link, I think I understand, that my proposal for an alternative definition isn't the right way to do it. Take for example the definition of a smooth morphism in Quing's book: Let $Y$ be locally noetherian and let $f:X \rightarrow Y$ be a morphism of finite type. [...]. We say that $f$ is smooth if [...]. Now, I would have changed that to "a morphism $f:X \rightarrow Y$ of schemes is called smooth if it is of finite type, if $Y$ is locally noetherian and if [...]". But due to the more general definition given in the stacks project, this wouldn't be a good

@Georges: The intention of the second point in my edit above was to make clear that I'm not criticizing Liu Quing, his book, or algebraic geometry in general, and that my questions just emerged from getting confused by some definitions and knowing about experts here who can explain how I have to deal with those. Getting down voted for asking questions about a mathematical textbook (let it be the best written book on earth; I am telling people to take a look at Quing's book instead of Hartshorne's for quite some time now by the way) is something that produces confusion, too.

@Wadim...or check Alan's answer.

@Wadim: I don't think the author meant "order" 3. There is only one group of order 3, namely Z/(3) which is pretty abelian...

mathdict.chitanka.info

I partially can. Szamuely discusses some aspects of Dedekind schemes which were quite enlightening for me. You can take a look at this chapter here: renyi.hu/~szamuely/gal6-7.pdf. Unfortunately I don't know about the rest of this book.

"Galois theories" by Francis Borceux and George Janelidze is quite nice. (books.google.com/…)

Could you perhaps add the definition of a complete group?