= Solution
For $p=3$, a <root of a polynomial>[root] would have <valuation> zero because five is a <unit in a ring>[unit]. It would therefore reduce to a <unit in a ring>[unit] cube modulo nine. The only cube <residue classes> modulo nine are $0,1,-1$, none equal to five. Thus \b[no <root of a polynomial>[root] exists in $\mathbb Q_3$].
For $p=5$, the <valuation> equation would be $3v_5(x)=1$, impossible because the <valuation> of a nonzero element of $\mathbb Q_5$ is an <integer>. Thus \b[no <root of a polynomial>[root] exists in $\mathbb Q_5$].
For $p=13$, $7^3\equiv5\pmod{13}$ and the <derivative> $3\cdot7^2\equiv4\not\equiv0\pmod{13}$. <Hensel's lemma> lifts this simple <root of a polynomial>[root] to a <root of a polynomial>[root] in $\mathbb Z_{13}\subseteq\mathbb Q_{13}$. Thus \b[a <root of a polynomial>[root] exists in $\mathbb Q_{13}$].
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