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.


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.

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  • $\begingroup$ 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

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