Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.)
$(1)$ If $M$ is a finite Galois extension of $F$ with Galois group $G$, then we have the canonical isomorphism $$M\otimes_F M \xrightarrow{\sim} M^{G}:(x,y)\mapsto (x\cdot g(y))_{g\in G}.$$
$(2)$ Suppose that $K,L \subseteq M$ are two finite extensions of $F$. If we write $L=F[x]/f(x)$, then $K\otimes_F L = K[x]/f(x)$ is a direct sum of finite extensions of $F$. For example, $K\otimes_F L = \bigoplus_i K_i$.
I am wondering:
(a) Can we describe $K\otimes_F L$ in a manner similar to $(1)$ ?
For example, let $H=\operatorname{Aut}(L/F)\subseteq G$, then it seems that $$M\otimes_F L \xrightarrow{\sim} M^{G/H}:(x,y)\mapsto (x\cdot g(y))_{g\in G/H}.$$
(b) For the case $F=\mathbb{Q}_p$, let $v_p(-)$ be the $p$-adic valuation, which is well-defined on finite extensions of $F$. Suppose that $x\in K$, $y\in L$, $x,y\neq 0$. Let $z$ be the summand of $x\otimes y$ in some $K_i$ in the right hand of $K\otimes_F L = \bigoplus_i K_i$. Do we have $v_p(x)+v_p(y) = v_p(z)$?
(c) Write $O_{\square}$ for the ring of integers of $\square\in\{F,K,L,M,K_i\}$. Can we describe the integral closure of $O_K\otimes_{O_F}O_L$ in $K\otimes L$? Does it equal (or is it a subset of) $\bigoplus_i O_{K_i}$?
