Here is a nifty sufficient but not necessary condition on whether a quintic is solvable in radicals or not. Given,
If there is an ordering of its roots such that,
or alternatively, its coefficients are related by,
then  is solvable as,
where the are the roots of the simple quartic,
Note that  in fact is the constant term of Cayley’s resolvent sextic and is only quadratic in f. Using another relation among the , Dummit’s resolvent has a constant term that is already a quartic in f, hence the choice of relation matters.
Example 1: This family of quintics by this author satisfies ,
Let n = 1, and we have,
Example 2: Another good example of  is Emma Lehmer’s quintic,
The linear transformation,
will reduce it into the form of , and it will then be seen its coefficients obey . As a particular example, let n = 5 and we have the reduced form,
Let its roots be,
and we find that indeed it obeys .
Unfortunately, no similar simple relation between the coefficients of a solvable septic, or 7th degree equation, is yet known.