The Stacks project assumes the reader knows quite a bit of field theory, including Galois theory. It might be a good idea to add a chapter on field theory leading up to and including Galois theory. Preferably this chapter would use as little commutative algebra as possible (so it could be placed before the commutative algebra chapter).

The sections Tag 030D, Tag 037H, and Tag 09DU would then be moved there. Currently this material is mostly just stated and probably in the wrong order too.

Perhaps more advanced material, e.g. on separable extensions of fields, would not be moved into the new chapter (although it is entirely possible that a more elementary treatment of separable extensions exists — certainly the current exposition isn’t optimal).

How can you help? Well, if you happen to have latexed (in your younger years) a writeup of field theory, then send it over to stacks.project@gmail.com and if suitable, we will distort it out of shape and put it in. (Important note: you cannot lose any reputation points by doing this, it is understood that you just send it in the hope that it will be useful.) If not you can just take one of the lemmas in one of the sections listed above and write up a proof and send that over. Thanks!

Update August 19, 2013: I went ahead and took the chapter “Fields and Extensions” (with permission) from the CRing project. Some edits needed to be made to adapt the style of the new chapter to the Stacks project. As I got further into it I changed more and more, due to personal preferences. But I hope some of the spirit of the original chapter still remains. Enjoy!

PS: Some statements are still given without proofs, in particular the statements on Kummer and Artin-Schreier extensions. It would be wonderful if somebody could add them.