= Quotient-ring proof of the Kummer-Dedekind theorem
{title2=$\mathcal O_K/p\mathcal O_K\simeq\mathbb F_p[X]/(\overline m)$}
For an integral generator $\alpha$ of a <number field>, let $A=\mathbb Z[\alpha]$ and suppose $p$ is coprime to $[\mathcal O_K:A]$. Localizing at $p$ makes $A$ and the <ring of integers> equal, so reduction gives the displayed <quotient ring>. If $\overline m=\prod g_i^{e_i}$ with distinct monic <irreducible polynomials>, the <Chinese remainder theorem> gives local factors $\mathbb F_p[X]/(g_i^{e_i})$. Their <residue fields> have degrees $\deg g_i$ and their <maximal ideals> have <nilpotence indices of an ideal> $e_i$. Localizing the <Dedekind domain> $\mathcal O_K$ at the corresponding <prime ideals> gives <discrete valuation rings>, where this same <nilpotence index of an ideal> is the <ramification index of a prime ideal>. Thus $(p)=\prod(p,g_i(\alpha))^{e_i}$ and each <residue degree> is $\deg g_i$.
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