= Solution
An element $x\in L$ is <separable algebraic element>[separable over] $K$ when its <minimal polynomial> $m_x(T)\in K[T]$ has distinct roots in a <splitting field>.
Because $m_x$ is irreducible, it is separable exactly when
$$
\gcd(m_x,m_x')=1.
$$
If $m_x'\ne0$, then $\deg m_x'<\deg m_x$, so irreducibility makes this greatest common divisor equal to one. If $m_x'=0$, the <polynomial with zero formal derivative in characteristic p>[zero-derivative criterion] gives
$$
m_x(T)=g(T^p)
$$
for some $g\in K[T]$, and $m_x$ is inseparable. Therefore
$$
\boxed{x\text{ is separable over }K
\iff m_x(T)\ne g(T^p)\text{ for every }g\in K[T].}
$$
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