What is the tensor product $M_n(L)\otimes_K L$, where $L/K$ is a quadratic extension?

Question

What is the tensor product $M_n(L)\otimes_K L$, where $L/K$ is a quadratic extension?

Let $K$ be a field of characteristic $0$, $L/K$ a quadratic extension. Let $\rho\in \operatorname{Gal}(L/K)$ denote the nontrivial element of the Galois group. Let $M_n(L)$ denote the algebra of $n\times n$ matrices over $L$. The "conjugation" $\rho$ acts on $M_n(L)$ by conjugating each matrix element.

I want to understand the tensor product $M_n(L)\otimes_K L$ as an algebra over the right-hand $L$, with conjugation coming from the action of $\rho$ on $M_n(L)$. Of course it must be $M_n(L)\oplus M_n(L)$ (and the conjugation must send $(X,Y)\in M_n(L)\oplus M_n(L)$ to $(Y^\rho,X^\rho)$). Why is it so, and how can I construct an explicit isomorphism of $L$-algebras $M_n(L)\otimes_K L\to M_n(L)\oplus M_n(L)$ (in light of the answers to my previous question Tensor product algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$)?

Answer 1

You may take the $\operatorname{Gal}(L/K)$-equivariant isomorphism of $L$-algebras given by $$M_n(L)\otimes_K L\xrightarrow {\cong } M_n(L)\times M_n(L):A\otimes l\mapsto (A\cdot l,A^\rho\cdot l) $$