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From Hilbert 90 (or, more precisely, a generalisation thereof), we know that the first (Galois) cohomology group of $GL_n$ is trivial, no matter the field of definition. However, for unitary groups, which are outer forms of the general linear group, the story is different. As I have heard again and again, the first cohomology group is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. Is there a quick proof? I suspect (based on almost nothing) that this has to do with the fact that a unitary group splits over a quadratic extension. Hence, the question is : could someone give a proof/reference of this fact?

Added: What I would be looking for is a way to explicitly describe the non-trivial cocycle in $H^1(F,U(p,q))$, if this is at all possible. In fact, to be more precise, I am looking at $U(n,n)$ over $F$, where $F$ is either a number field or a non-archimedean local field.

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A very general version of this is in Platonov/Rapinchuk, "Algebraic Groups and Number Theory", Proposition 2.16 (page 86). Unfortunately, it looks like it requires a bit of work to extract the statement you want from the statement they give... It's probably also in "The Book of Involutions", but I don't have it on hand. – B R Apr 5 '12 at 2:11
I should really get my hands on Platonov/Rapinchuk. Thank you (again) for this reference! As for the book of involutions, a quick glance at the table of contents shows there might be something interesting in Chapter 7. I'll have a look at this one too. – M Turgeon Apr 5 '12 at 12:06
@BR Thank you for the reference, it is a wonderful book. But, as my edited question now says, I would like a way to describe the non-trivial cycle, and Proposition 2.16. is far from giving a way to do this. – M Turgeon May 2 '12 at 21:02
I wish I could provide more help! Maybe ask at MO? – B R May 2 '12 at 22:45

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