I would like to find a polynomial in $\Bbb Q[X]$ whose splitting field has Galois group $\Bbb Z_6$.
Since $\Bbb Z_6$ can be embedded in $S_n$ only when $n \ge 5$, the degree of the polynomial must be $\ge 5$.
Since $\Bbb Z_6 \cong \Bbb Z_2 \times \Bbb Z_3$, our polynomial is reducible.
Aiming for a minimal example, my polynomial would have degree exactly $5$ and would reduce into two irreducible polynomials, one with degree $2$ and one with degree $3$.
To prevent the one with degree $3$ from contributing $S_3$ instead of $\Bbb Z_3$, its discriminant must be a rational square, so it may for example be $x^3-3x+1$. It is irreducible by rational root theorem.
Our factor of degree $2$ could be $x^2-2$.
Therefore, our polynomial is $(x^2-2)(x^3-3x+1) = x^5-5x^3+x^2+6x-2$.
Am I correct?