In ch. I, §8, p. 49 of the English translation of $\textit{Algebraic Number Theory}$, Neukirch states the following (I paraphrase, but retain his notation):

Let $L/K$ be an extension of number fields, and let $\mathcal{O}$ and $\mathcal{o}$ be their respective rings of integers.

Then every prime ideal $\mathfrak{p} \subset \mathcal{o}$ possesses a unique factorisation

$$\mathfrak{p} = \mathfrak{P}_1^{e_1}...\mathfrak{P}_1^{e_1}$$

where the $\mathfrak{P}_i$ are prime ideals in $\mathcal{O}$.

He then says that the prime ideal $\mathfrak{P}_i$ in the above factorisation is said to be $\textbf{unramified}$ over $K$ if $e_i = 1$ ($\textit{i.e.}$ if the prime only appears once in the factorisation) $\underline{\textit{and}}$ the residue field extension $(\mathcal{O}/\mathfrak{P}_i)/(\mathcal{o}/\mathfrak{p})$ is separable.

But $(\mathcal{O}/\mathfrak{P}_i)/(\mathcal{o}/\mathfrak{p})$ is a finite extension of finite fields, so will it not always be separable? Why is this included in the definition?

Below I have included the excerpt from the book I am referring to.

Ramified primes

Thank you for your attention.

$\textbf{Addendum:}$ In response to the comments below, include two more excerpts from ch. I, §9, p. 58 and p. 59 respectively.

From p. 58:

Ramified primes2

From p. 59:

Ramified primes3

I should explain that his notation is $\kappa(\mathfrak{P}) := \mathcal{O}/\mathfrak{P}$ and $\kappa(\mathfrak{p}) := \mathcal{o}/\mathfrak{p}$.

Now if the assumption was only included for formal reasons (because he did not want to assume that finite fields are perfect), then why would he repeatedly refer to the separability of the residue field extension as a "special case"?

  • 1
    $\begingroup$ There are examples of (things also called) ramification in some other contexts for which separability is nontrivial, but it is automatic for the case here. Maybe Neukirch didn't want to assume the result that finite fields are perfect? (I don't have my copy at hand to check, but it would seem like he uses enough Galois theory to prove or assume that, though.) $\endgroup$ – anomaly Feb 26 at 19:59
  • 2
    $\begingroup$ It's also possible he wrote it out of habit, and kind of momentarily forgot it was automatic for number fields. $\endgroup$ – Captain Lama Feb 26 at 20:03
  • $\begingroup$ @anomaly That sounds plausible, but then why would he repeatedly refer to it as a "special case"? See the bit added above. $\endgroup$ – Heinrich Wagner Feb 26 at 20:21

I took a look at the corresponding section in the book. Although the end goal is of course to apply all this to number fields, this is actually not the assumption in that section. Neukirch works at the level of generality of Dedekind domain, not rings of integers of number fields. And in that more general case, the residue fields don't have to be finite, and the inertia extension does not have to be separable.


Given your interest in L-functions you should look at the finite extensions of the $p$-adic integers $\Bbb{Z}_p$. The main point is Hensel lemma: if $f\in \Bbb{Z}_p[x]$ and $f(a)=0\bmod p,f'(a)\ne 0\bmod p$ then there is $b\in \Bbb{Z}_p$ such that $b=a\bmod p, f(b)=0$. This is what makes $p$-adic fields much simpler than $\Bbb{Q}$, because we can use this to classify all the extensions and connect some apparently very different fields. In the general setting we replace $\Bbb{Z}_p$ by $\varprojlim O/\mathfrak{p}^n$ where $\mathfrak{p}$ is a non-zero invertible prime ideal of $O$ (corresponding to a discrete valuation). The unramified condition is what we need to ensure the extension is generated by the lift of some roots of separable polynomials $\bmod \mathfrak{p}$. And we can understand $O$ by looking at $\varprojlim O/\mathfrak{p}^n$ for each prime ideal.


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