[[Splitting of prime ideals in a number field]]
# Kummer's factorization theorem

Let $K = \mathbb{Q}(\vartheta)$ be a [[number field]] where $\vartheta$ is an [[algebraic integer]], 
and suppose $p$ is a prime number not dividing the [[annoying index]]
 $\abs{ \mathcal{O}_{K} / \mathbb{Z}[\vartheta]}$.
 Let $m_{\vartheta}(x) \in \mathbb{Z}[x]$ be the [[Algebraic element|minimal polynomial]] of $\vartheta$, and write
 $$
\begin{align*}
m_{\vartheta}(x) \equiv_{p} \prod_{i=1}^g f_{i}(x)^{e_{i}}.
\end{align*}
$$
for $f_{i}(x) \in \mathbb{Z}[x]$ [[Polynomial ring#^irreducible]] mod $p$.
Then 
$$
\begin{align*}
p \mathcal{O}_{K} = \prod_{i=1}^g \mathfrak{p}_{i}^{e_{i}}
\end{align*}
$$
where $\mathfrak{p}_{i} = \langle p, f_{i}(\vartheta) \rangle$ are distinct [[Prime ideal|prime ideals]] of norm $\opn N(\mathfrak{p}_{i}) = p^{\deg f_{i}}$. #m/thm/num/alg 
We also have
$$
\begin{align*}
\sum_{i=1}^g e_{i} \deg f_{i} = n
\end{align*}
$$

> [!missing]- Proof
> #missing/proof

## Corollaries

See [[Splitting of prime ideals in a number field]].


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