I wish to determine if $f(x)=x^5+x^4+x^3-2x^2-2x+5$ is solvable by radicals over $\mathbb{Q}$. In other words, I want to know if its Galois group is solvable.
I haven't gotten anywhere trying to find the roots explicitly. Also, the result for polynomials with exactly three real roots doesn't apply here. How should I approach this problem?
According to a paper by Dummit, "SOLVING SOLVABLE QUINTICS", a quintic is solvable by radicals iff its Galois group is is contained in the Frobenius group $F_{20}$ of order $20$ in the symmetric group $S_5$. An algorithm and formula is provided as well.
Abstract:
It is well–known that an irreducible quintic with coefficients in the rational numbers $\mathbb{Q}$ is solvable by radicals if and only if its Galois group is contained in the Frobenius group $F_{20}$ of order $20$, i.e. if and only if the Galois group is isomorphic to $F_{20}$, to the dihedral group $D_{10}$ of order $10$, or to the cyclic group $\mathbb{Z}/5\mathbb{Z}$. (More generally, for any prime $p$, it is easy to see that a solvable subgroup of the symmetric group $S_p$ whose order is divisible by $p$ is contained in the normalizer of a Sylow $p$–subgroup of $S_p$.) The purpose here is to give a criterion for the solvability of such a quintic in terms of the existence of a rational root of an explicit associated resolvent sextic polynomial, and when this is the case, to give formulas for the roots analogous to Cardano’s formulas for the general cubic and quartic polynomials and to determine the precise Galois group. In particular, the roots are produced in an order which is a cyclic permutation of the roots, which can be useful in other computations. We work over the rationals $\mathbb{Q}$, but the results are valid over any field $K$ of characteristic different from $2$ and $5$.
The only general method for determining the number of real roots of a polynomial without solving the equation is the Sturm's theorem.
Its application is rather laborious, since require the calculus of a sequences of polynomials and the evaluation of table of their changes of sign in the interval in which we search the roots.
There is also a variant of the theorem ( proved by J.M.Thomas), that allows to determine the multiplicity of the roots. You can find some examples of use of this method in : B.E. Meserve, Fundamental Concepts of Algebra, pag. 160...
This polynomial has Galois group $S_5$ (this can be shown using a theorem of Frobenius by reducing modulo primes and considering the irreducible factors), so it is not solvable by radicals.
