Local factorization of X3 plus 25X2 minus 50X plus 40
= Local factorization of X3 plus 25X2 minus 50X plus 40
{title2=$X^3+25X^2-50X+40$ over $\mathbb Q_2$ and $\mathbb Q_5$}
The polynomial
$$
F(X)=X^3+25X^2-50X+40
$$
splits into three linear factors over $\mathbb Q_2$. Over $\mathbb Q_5$ it is an <Eisenstein polynomial>, so adjoining a root gives a totally ramified cubic extension. Its splitting field over $\mathbb Q_5$ has <decomposition group> $S_3$ and <inertia group> $A_3\cong C_3$.