Cubic field of discriminant minus 307
= Cubic field of discriminant minus 307
{title2=$K=\mathbb Q(\alpha),\ \alpha^3-5\alpha+8=0$}
The polynomial $X^3-5X+8$ is irreducible over $\mathbb Q$. For
$$
\beta=\frac{\alpha+\alpha^2}{2},
$$
one has
$$
\beta^3-5\beta^2+11\beta-12=0,
\qquad
\alpha=-4+3\beta-\beta^2.
$$
The field has ring of integers $\mathbb Z[\beta]$ and field discriminant $-307$.