= Equal absolute values of algebraic conjugates
{title2=$|\alpha_i|=|\alpha_j|$}
Suppose a <Non-Archimedean absolute value> on $K$ has a unique extension to every finite <field extension>. Any K-isomorphism $K(\alpha_i)\to K(\alpha_j)$ between conjugate roots pulls the absolute value back to an extension on the source field, which must be the given one. Thus $|h(\alpha_i)|=|h(\alpha_j)|$ for every $h\in K[X]$. This argument includes inseparable <irreducible polynomials>: no transitive Galois action or completeness assumption is needed.
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