 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[No prime works.] The <P-adic valuation> induces a well-defined <homomorphism>
$$
\mathbb Q_p^*/(\mathbb Q_p^*)^p\longrightarrow\mathbb Z/p\mathbb Z,
\qquad[a]\longmapsto v_p(a)\bmod p.
$$
The class of $p$ maps to one, whereas the class of $p^p$ maps to zero. They therefore cannot be equal. Equivalently, $p$ cannot be a $p$th power because its valuation is not divisible by $p$.

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