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Wadim, it looks to me that the denominator $n+1$ here should be $n-1$ to match the recurrence relation (2.4) from your paper. I don't know if that matters much though.
@Victor: I was inclined to say the same after I first read this, but then I looked again into the part on external Zappa-Szep products, and I should admit that in a sense this is a construction in the same way the semi-direct product is a construction. The semi-direct product depends on some data (action of H on K); similarly, this product depends on the mappings $\alpha$ and $\beta$ satisfying compatibility condition...
Then I'd strongly recommend you to rewrite the title of the question. Maybe "How to prove that Br(F(t))=0 without Tsen's theorem?", or something like that. This way you stand a better chance to attract the attention of someone who actually knows the answer...
I think "stars and bars" is a good term, - I've never heard of it before, but I was able to guess immediately what it was referring to among things I know, and that's a sign of a very good term!
Nice question!!! A small remark on "this is called the density theorem, I believe" - I think in the case of finite groups it is called Burnside theorem (one of many theorems of Burnside!), while you are right that it immediately follows from Jacobson's density theorem...
