Is there a Cayley resolvent for sextic polynomials

galois-theoryreference-request

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?

Best Answer

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.

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Is this sextic equation solvable

You can compute Galois groups of polynomials using the open-source computer algebra system Sage. The documentation for Sage's tools for working with Galois groups can be found here. To give an example:

A.<x> = QQ[]   # defines A = Q[x] to be the polynomial ring in x over the rationals
poly = x^6 + 3*x^5 + 3*x^4 + 3*x^3 + 2*x^2 + 1
G = poly.galois_group()
G.is_solvable()

If you want to do the computation by hand (or understand how computer implementations work), the book A Course in Computational Number Theory by Cohen (Graduate Texts in Mathematics 138, Springer-Verlag, 1993) is a good reference for efficient algorithms. In particular, section 6.3.5 deals with the case of determining the Galois group of a sextic polynomial. The method described there uses resolvent polynomials to distinguish between the 16 possible transitive subgroups of $S_6$.

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Is this sextic equation solvable

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