I'm trying to figure out if these statements are equivalent for an arbitrary field extension $L/k$, such that $\text{char} (k) = p$ and $A$ is a reduced commutative $k$-algebra.
$1)$ $L/k$ is separable (it has a separating transcendence basis).
$2)$ $L \otimes_k k^{p^{-1}}$ is reduced, an integral domain or a field.
$3)$ $L \otimes_k k^{p^{-r}}$ is reduced, an integral domain or a field for each $r = 0, 1, ..., \infty$.
$4)$ $A \otimes_k L$ is reduced.
$5)$ $A \otimes_k L$ is semisimple.
$6)$ $L\otimes_k F$ is reduced for all field extensions $F$ of $k$.
$7)$ $L \otimes_k F$ is semisimple for all field extensions $F$ of $k$.
I proved $1)$ implies $3)$. Furthermore, in $4$ and $5$, I'm trying to make an analogy between perfect fields and separable field extensions.
Thanks in advance.