6
$\begingroup$

Let $K$ be an algebraic number field, i.e. a finite field extension of $\Bbb{Q}$. I would like to prove that every non-archimedean absolute value on $K$ is equivalent to $$ |x|_{\mathfrak{p}} := N_K(\mathfrak{p})^{-\text{ord}_{\mathfrak{p}}(x)} $$ for some prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$, where $|0|_\mathfrak{p} = 0$ by convention and otherwise $\text{ord}_{\mathfrak{p}}(x)$ is the exponent of $\mathfrak{p}$ in the prime factorization of $(x)$, and where $N_K(\mathfrak{p}) = |\mathcal{O}_K/\mathfrak{p}|$ is the absolute norm of $\mathfrak{p}$.

Preliminary question: Is this actually true?

If so, how can I prove this? I know that the completion of $K$ with respect to a non-archimedean absolute value is isomorphic to a finite extension of $\Bbb{Q}_p$ for some prime number $p$, and that $$ |x|_p = |N_{L/\Bbb{Q}_p}(x)|_p^{1/[L:\Bbb{Q}_p]} $$ is the unique extension of $|\cdot|_p$ to a finite extension $L$ of $\Bbb{Q}_p$, but I don't know how to conclude from here.

See also this related question.

$\endgroup$
4
$\begingroup$

Yes, this is true. The case for $\mathbb{Q}$ is sometimes called Ostrowski's Theorem, although related results also go by this name. The proof for number fields is extremely similar (and sometimes also called Ostrowski's Theorem).

A complete proof can be found here. It is not very long.


Some additional (more reputable) references per request:

In Chapter 2.1 of Silverman's The Arithmetic of Dynamical Systems, Ostrowski's Theorem is given. [Incidentally, I learned this fact from him]. But it's given without proof. For proofs, he references Theorem 3, 1.4.2, of Number Theory by Borevich and Shafarevich or Theorem 1, I.2, of Koblitz's p-adic Numbers book. I haven't read either of these, but I do not doubt Silverman.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks. Do you know of a textbook or another resource that I can cite which contains that proof? $\endgroup$ – A.P. Jun 19 '15 at 9:10
  • $\begingroup$ I've updated the answer to include some hardened references that I know $\endgroup$ – davidlowryduda Jun 19 '15 at 9:31
  • $\begingroup$ I checked those references. Sadly, (just like every other reference I was able to find) they mention only the version of Ostrowski's theorem for $\Bbb{Q}$. I don't see any immediate way to derive the version for $K \supset \Bbb{Q}$ from that one, though. $\endgroup$ – A.P. Jun 19 '15 at 10:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.