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Suppose $n\geq 5$ and $f(x)$ is a polynomial of degree $n$, then in general $f(x)$ is not solvable by radicals, but there are certain special polynomials which are solvable by radicals. Is there a characterization of all such polynomials ?

Edit: Of course Galois group is solvable is one characterization, but I am looking for more stronger charaterization for some special values of $n$ (say $n=5$, or $n$ is prime), if not for general $n$.

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Of course. The Galois group f the polynomial must be solvable (in the sense of group theory). This is completely standard and can be found in any advanced algebra textbook, so this question is not appropriate for MO, which is devoted to research-level questions. Please see the FAQ to find a list of forums that could help you. – Francesco Polizzi Sep 6 at 6:31
Please see the edit, I am interested in some simple characterization for some special values of $n$ if possible. – pritam Sep 6 at 6:59
Are you asking for a characterization of solvable transitive subgroups of $S_n$, or an easily computable condition on polynomials? – S. Carnahan♦ Sep 6 at 7:32
Yes, an easily computable condition on polynomials – pritam Sep 6 at 7:45
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 In en.wikipedia.org/wiki/Quintic_function you can find some of these conditions for the quintic polynomial. However I do not know how much they are "easily computable" – Francesco Polizzi Sep 6 at 7:57
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closed as too localized by Francesco Polizzi, Andy Putman, Chris Godsil, DamienC, Fernando Muro Sep 6 at 8:12

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