Here is a nifty *sufficient but not necessary* condition on whether a quintic is solvable in radicals or not. Given,

[1]

If there is an ordering of its roots such that,

[2]

or alternatively, its coefficients are related by,

[3]

then [1] is solvable as,

where the are the roots of the simple quartic,

Note that [3] 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 [3],

Let *n* = 1, and we have,

Quartic is,

such that,

Example 2: Another good example of [3] is *Emma Lehmer’s quintic*,

The linear transformation,

will reduce it into the form of [1], and it will then be seen its coefficients obey [3]. As a particular example, let *n* = 5 and we have the reduced form,

Let its roots be,

and we find that indeed it obeys [2].

Unfortunately, no similar simple relation between the coefficients of a solvable septic, or 7th degree equation, is yet known.

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Posted by kadir saim uyanık (male, 49) on October 25, 2013 at 7:02 am

Thank you very much for your valuable contributions for solvable quintics. I wish you will reach new findings and share them with us. Yours sincerely.