= The Eisenstein prime does not divide the power-basis index
{c}
Let $f=X^n+a_{n-1}X^{n-1}+\cdots+a_0\in\mathbb Z[X]$ satisfy <Eisenstein's criterion> at a <prime number> $p$, and let $\theta$ be a root. Then
$$
p\nmid[\mathcal O_{\mathbb Q(\theta)}:\mathbb Z[\theta]].
$$
Indeed, $\theta^n/p$ is an <algebraic integer>, whereas for $p\nmid r$ and $0\le i<n$ the <field norm> of $r\theta^i/p$ has $p$-adic valuation $i-n<0$, so that element is not integral. If $\gamma=p^{-1}\sum_{i=0}^{n-1}r_i\theta^i$ is integral, multiply it by $\theta^{n-1}$. Every term except $r_0\theta^{n-1}/p$ is integral; thus $p\mid r_0$. Subtract $r_0/p$, multiply the remainder by $\theta^{n-2}$, and continue to obtain $p\mid r_i$ for every $i$. Hence there is no element of order $p$ in the finite group $\mathcal O_{\mathbb Q(\theta)}/\mathbb Z[\theta]$, proving the claim. The result concerns the Eisenstein prime only: other primes may still divide the index.
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