In our abstract algebra class, we had two lectures on field extensions. Then we were given the following homework problem (with this problem on diagonal $K$-algebra).
Let $K$ be a field. A commutative $K$-algebra $A$ such that $[A:K]$ is finite and $A$ is isomorphic to $K^X$ where $|X|=n$ is called a diagonal $K$-algebra.
A commutative $K$-algebra $A$ is said to be potentially diagonal if there exists an extension $L$ of $K$ such that $L \otimes_K A$ is a diagonal $L$-algebra.
There are two questions.
1) Let $\Omega$ be an algebraic closure of $K$. Let $P(X) \neq 0$ be a polynomial of degree $n$ in $K[X]$, and let $A = K[X]/(P(X))$. Assume $P(X)$ has $n$ distinct roots in $\Omega$. Show that $A$ is potentially diagonal.
2) Let $B$ and $C$ be potentially diagonal algebras, and let $A = B \otimes_K C$. Show that $A$ is potentially diagonal.
I am really lost as to how to solve these two questions. Any help/ hint/ comment is very much appreciated.
For 1)
Show that $\Omega\otimes_KA$ is isomorphic to the $\Omega$-algebra $\Omega[X]/(P(X))$.
Since $P$ has no repeated factor, it admits a factorization $P(X)=\prod_{i=1}^n(X-\omega_i)$ with the $\omega_1,\dots,\omega_n$ are pairwise distinct elements of $\Omega$. Use the Chinese remainder theorem to show that $\Omega[X]/(P(X))$ is isomorphic to the diagonal algebra $\Omega^n$.
For 2)
Show that $A$ is a potentially diagonal $K$-algebra if and only if $\Omega\otimes_KA$ is a diagonal algebra.
Show that $\Omega\otimes_K(A\otimes_KB)$ is isomorphic to $(\Omega\otimes A)\otimes_\Omega(\Omega\otimes_KB)$.
Use these two facts to conclude that the tensor product of two potentially diagonal algebras is potentially diagonal.