I was looking over the notes and saw a few items omitted by Milne.

First, in the proof of Hilbert’s theorem 90, there is no proof of the linear independence of distinct characters on a group. You can try to figure it out yourself (quite easy) or look at the notes with that title Keith Conrad’s webpage. (For some strange reason, I can’t create a direct link to the paper.)

Secondly, Milne has a cryptic remark without definition or proof about a ‘Verlagerung’ (transfer map) in Proposition 3.2. This is reference to the fact that the restriction map for Tate cohomology in degree minus 2 corresponds to a classical formula for a map between abelianizations. This proof is not so straightforward, and can be found in Prop.11.12 of these nice notes by Holden Lee. It seems they’re from a reading course he took with Professor Shin Sug Woo.

If some of you would like to read a more leisurely exposition of group cohomology, I found online these notes by Kenneth Brown.

Finally, this old paper of Oort contains a discussion of extensions and cup products.