actually I would be really thankful for a literature reference because it is very unclear to me how Japanese enters this problem. S_2 and separatedness seem necessary. But where would I have to take an integral closure?

(2) The statement "Gorenstein implies Cohen-Macaulay implies S2. So the statement is also true for hypersurfaces and complete intersections, which could be very singular and non-reduced." is a little dangerous. So, personally I don't know how Japanese enters the proof - but irrespective of that lack of understanding - normal schemes, Gorenstein schemes etc. need not be Japanese. Varieties are (in all standard definitions of variety), but varieties are usually also assumed to be reduced. So a lot of care is needed here

I think there are some slight imprecisions in this answer: (1) a coherent sheaf will NOT have a unique extension. The uniqueness is only possible for vector bundles. For example, take the pushforward of the structure sheaf of the reduced cl. subscheme X-U. It is coherent, and an extension of the zero sheaf. But the zero sheaf is extended by zero as well. So there is no uniqueness of the extension for coh. sheaves

very fair point. I was blind, it doesn't work :-( Well, but thanks alot for pointing this out.

you CAN talk about roots without convergence matters since you may use Weierstrass preparation to reduce your question to a polynomial times some unit power series. Take the roots to be those of the polynomial. Similarly the Galois action sends units to units, so the Galois action is fine as well.