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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.

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    @awllower: I mean the notion is not invariant under, for example, replacing $p(x)$ with $p(x+a)$ where $a \in F$.2011-11-16

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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 $p$.