Is there a Cayley resolvent for sextic polynomials?

I have no Galois Theory knowledge and a specific sextic polynomial that I'd like to prove has roots that are inexpressible with radicals. Putting it into Wolfram Alpha yields only approximations, but I'm looking for a more rigorous argument than this.

Searching on Wikipedia, I found out about Cayley's Resolvent, which provides a nice deterministic test for when the roots of a quintic are expresssible with radicals. Is there an analogue for sextics (or even higher dimensions)? If so, could someone point me towards a paper or book where this test is proven (or even just reputably claimed) to work, for the purpose of citation?

Solution 1:

Thomas Hagedorn's 1999 paper General Formulas for Solvable Sextic Equations uses two resolvents of degrees $10$ and $15$, denoted $f_{10}$ and $f_{15}$ respectively – necessary because two groups of orders $48$ and $72$ are needed to exactly cover all transitive solvable groups on $6$ elements as subgroups – to determine solvability of a sextic. At least one of the following has to hold:

• $f_{10}$ has a rational root
• $f_{15}$ has a rational root of multiplicity not $5$
• $f_{15}$ has a rational root of multiplicity $5$ and $f_{10}$'s $\mathbb Q$-irreducible factorisation has degrees $4,6$

If the Galois group is contained in the order-$72$ group there is a quadratic extension over which the sextic splits into two cubics. If contained in the order-$48$ group there is a cubic extension over which the sextic splits into a quadratic and quartic.