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I ha come across a book on groups as Galois groups, and in the introduction it mentions the paper by I.R. Shafarevich which says that every solvable group can be realized as Galois groups of some Galois extensions over $\mathbb{Q}$. It, however, does not tell us where to find that paper; therefore, I would like to know where I can find it.
In any case, thanks very much.

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up vote 5 down vote accepted

The original paper, in Russian, was:

  • I.R. Shafarevich, Construction of fields of algebraic numbers with given solvable Galois group (russian) Izv. Akad. Nauk SSSR, Ser. Mat. 18 (1954), 525–578. English translation in Amer. Math. Soc. Transl. 4 (1956), 185-237.

The original paper had an error in it, and there was a correction appended in 1989. A complete proof can be found in

  • J. Neukirch, A. Schmidt & K. Wingberg, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag, 2000.

There is also a proof available here, in a paper by Alexander Schmidt and Kay Wingberg. The paper is available in Postscript. will create a PDF from an uploaded PS file, so it can be used to convert it if you cannot open PS documents.

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Since I cannot open the files, do you have any other ways in reading the proof? Thanks. – awllower Mar 6 '11 at 3:55
And it is pretty helpful, while it will be more helpful if any one puts it community wiki, thanks. – awllower Mar 6 '11 at 3:56
@awllower: The file is in postscript. I was able to create a PDF of it by downloading the file, and then going to to create a PDF file. Will that do? – Arturo Magidin Mar 6 '11 at 3:59
@awllower: I don't understand your second comment. – Arturo Magidin Mar 6 '11 at 3:59
This is already there in Arturo's answer, but is perhaps worth stressing even more: there is a notorious flaw in Shafarevich's original argument "with respect to the prime $2$". For a while it was sort of folkloric that it could be fixed up, but there was no formal corrigendum (AFAIK). For these reasons as well as others I think this is a case where it would not be so rewarding to read the original paper. The book by Neukirch et al. that Arturo mentions is extremely well written and reliable: I recommend looking there for the proof. – Pete L. Clark Mar 6 '11 at 6:05

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