 Crossed-product algebra of a Galois extension

For a finite Galois extension $L/k$, Galois group $G$, and normalized two-cocycle $\varphi :G\times G\to L^{\times }$, the crossed-product algebra iswith $u_{\sigma }a=\sigma (a)u_{\sigma }$ and $u_{\sigma }{u}_{\tau }=\varphi (\sigma ,\tau )u_{\sigma \tau }$. It is a central simple $k$-algebra split by $L$.