$L/K$ is a finite Galois extension with group $G$. Let $F \in K[X]$ a separable monic polynomial of degree $n \geq 1$ and consider the $L$-algebra $B=L[X]/\langle F \rangle$. Coefficients of $F$ are in $K$ and we can define a natural action of $G$ on $B$. Is it true that we have:
\begin{equation} H^1(G,B^\times)=\{1\} \end{equation}
where $B^\times$ is the group of inversibles of $B$. Is there an elementary proof of that result ?
NB: if $F=X$ then $B=L$ and we have the Hilbert 90 classical theorem.