Is there a specific name for this notion of extensions of fields?

Question

As this question suggests, I quite like the notion of permuting the coefficients of polynomials.
And, moreover, I have another question on this direction:If L|F is a finite normal field extension, then it must be normal, then my question is: is there a name for the field extension L|F such that L contains all roots of polynomials obtained by permuting the coefficients of p(x) where p(x) is a polynomial one of whose roots lie in L.
In any case, thanks for paying attention.

Answer 1

Such a field is algebraically closed! Let $p \in K[x]$ be any polynomial, and consider $xp$; then $xp$ has a root in $K$ (namely, zero); so all the roots of $xp$ lie in $K$, and that includes all roots of $p$.