= Local prime decomposition of T5 plus 6T3 plus 252T2 plus 126
{title2=$T^5+6T^3+252T^2+126$}
This polynomial is <Eisenstein> at $2$. Over $\mathbb Q_3$, its <Newton polygon> has a single slope $-2/5$, so it is irreducible and a root generates a totally ramified extension of degree five. Over $\mathbb Q_7$, its factor degrees are $1,1,3$: the simple residue roots $1,-1$ lift, and the remaining roots have valuation $1/3$. Thus the degree-five <number field> generated by a root has one prime above $3$ and three above $7$.
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