= Ramification forced by a coprime root valuation
{title2=$\alpha^m=a,\quad\gcd(m,v_p(a))=1\implies m\mid e_{\mathfrak p/p}$}
Suppose $\alpha^m=a\in\mathbb Q^\times$ in a <number field> and $\mathfrak p$ lies above $p$. Its normalized integral <valuation> satisfies $m v_{\mathfrak p}(\alpha)=e_{\mathfrak p/p}v_p(a)$. If $m$ and $v_p(a)$ are coprime, then $m\mid e_{\mathfrak p/p}$. In a degree-$m$ field the <fundamental identity for prime decomposition> forces a unique prime, <ramification index> $m$, and <residue degree> one. This detects total <ramification of a prime> even when the displayed defining polynomial is not <Eisenstein> at $p$.
// Target: algebra.bigb
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