Let $K$ be an algebraic number field, and suppose its ring of integers is $\mathcal{O}_K = \mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}_K$. Let $f \in \mathbb{Z}[X]$ be the minimal polynomial of $\theta$, and suppose $\overline{f} = \overline{f_1}^{e_1} \cdots \overline{f_r}^{e_r} \in \mathbb{F}_p [X]$, where $p$ is a rational prime and $\overline{f_1}, \ldots, \overline{f_r}$ are distinct monic irreducible polynomials. Then Dedekind's theorem on the factorisation of rational primes tells us that the ideal $(p) \triangleleft \mathcal{O}_K$ factors as ${\mathfrak{p}_1}^{e_1} \cdots {\mathfrak{p}_r}^{e_r}$, where $\mathfrak{p}_1, \ldots, \mathfrak{p}_r$ are distinct prime ideals and moreover if $f_j \in \mathbb{Z}[X]$ is monic and reduces to $\overline{f_j} \in \mathbb{F}_p[X]$, then $\mathfrak{p}_j = (p, f_j(\theta))$.
In Lang's Algebraic Number Theory (1968, p.28) there is a proof of this theorem, which I can follow right up to the penultimate line. Then there's a mysterious equation which seems to come from nowhere, and it's literally the only obstruction to my understanding of the proof. The gist of the proof is as follows:
- First we construct the prime ideals $\mathfrak{p}_j$ as the kernels of homomorphisms $\mathbb{Z}[\theta] \to \mathbb{F}_p(\alpha)$, where $\theta$ is mapped to a root $\alpha$ of $\overline{f_j}$.
- Then, we show that $\mathfrak{p}_j = (p, f_j(\theta))$.
- Suppose $(p) = {\mathfrak{p}_1}^{e'_1} \cdots {\mathfrak{p}_r}^{e'_r}$. (I presume this is by unique prime factorisation in Dedekind domains? But then how do we know there aren't additional factors, which aren't the $\mathfrak{p}_j$ we've constructed?) Then, we note that $f – {f_1}^{e_1} \cdots {f_r}^{e_r} \in p \mathbb{Z}[X]$, so $f_1(\theta)^{e_1} \cdots f_r(\theta)^{e_r} \in (p)$.
- So it follows that ${\mathfrak{p}_1}^{e_1} \cdots {\mathfrak{p}_r}^{e_r} \subseteq (p) = {\mathfrak{p}_1}^{e'_1} \cdots {\mathfrak{p}_r}^{e'_r}$, and thus $e_j \le e'_j$.
- Let $d_j = \operatorname{deg} f_j$. Then clearly $n = \operatorname{deg} f = d_1 e_1 + \cdots d_r e_r$. I can see that if we can show that $d_1 e'_1 + \cdots + d_r e'_r = n$ then we are done. But why is the equation true? Does it follows from unique factorisation?
It feels like the proof shouldn't be this long though, and the Chinese remainder theorem doesn't seem to be invoked anywhere even though it seems to be the obvious thing to do. Is there a simpler proof, using the Chinese remainder theorem?
Best Answer
HINT $\rm\ \ e_i \le\: e_i'\ $ and $\rm \ d_1\ e_1 +\:\cdots\:+ d_r\ e_r\ =\ d_1\ e_1'+\:\cdots\:d_r\ e_r'\ \Rightarrow\ e_i = e_i'\:.$
Edit $\ $ To answer the clarified question, the equation $\rm\:n\ =\ \sum d_i e_i'$ follows from Proposition 21, namely $\rm\:n\ =\ [L,K]\ =\ \sum_{P|p}\ e_P\:f_P\:.$