Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup K$ acts on the $L$-algebra $L\otimes_K B$ (although not by $L$-linear homomorphisms), thus also on its unit group $\left(L\otimes_K B\right)^{\times}$. Is it true that $H^1\left(G,\left(L\otimes_K B\right)^{\times}\right)$ is the one-element set?
For $B=K$, this would be Hilbert 90. More generally, for $B$ being a matrix algebra over $K$, this would be an extension of Hilbert 90 Milne claims to hold in his CFT. My main reason for generalizing to arbitrary $B$ is to prove the following fact, known as Noether-Deuring theorem:
Let $A$ be a $K$-algebra, and let $U$ and $V$ be two finite-dimensional representations of $A$ over $K$. Then, $U$ and $V$ are isomorphic representations if and only if the representations $L\otimes_K U$ and $L\otimes_K V$ are isomorphic representations of the algebra $L\otimes_K A$. Note that this holds not only for Galois extensions $L\diagup K$ but for arbitrary field extensions $L\diagup K$, but the (Galois) case of finite fields is the hardest. This generalizes the fact that two matrices over some field are similar if and only if they are similar over a field extension, which, in turn, is a particular case of "Conjugacy rank" of two matrices over field extension .
