The author of my textbook argues that the quintic polynomial $3x^5-15x+5$ is not solvable by radicals over $\mathbb{Q}$ by showing that the Galois group of $3x^5-15x+5$ over $\mathbb{Q}$ is isomorphic to $S_{5}$ (which is not solvable).

But the argument given would seemingly apply to any quintic polynomial with integer coefficients that is irreducible over $\mathbb{Q}$ and has 3 distinct real roots and 2 non-real complex roots.

Are all quintic polynomials of this type not solvable by radicals?


Yes, if $f$ is irreducible, then $5\mid [K:\mathbb{Q}]$ where $K$ is the splitting field of $f$. Hence, the Galois group $G$ of $f$ contains a 5-cycle. Furthermore, if it has only two non-real roots, then $G$ also contains a transposition.

The result now follows from the fact that if $G<S_p$ (where $p$ is prime) is a subgroup that contains a $p$-cycle and a transposition, then $G=S_p$.

So not only is such a polynomial not solvable, its Galois group is $S_5$.


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