0
$\begingroup$

Let $k$ be a number field and $M_k$ be the set of standard absolute value on $k$, which means for $v \in M_k$, $v$ is an absolute value on $k$ whose restriction to $\mathbb{Q}$ is $|\cdot|_p$ where $p$ is a prime number or $\infty$.

I want to know any non-trivial example of $M_k$ or those elements. For example, let $k=\mathbb{Q}(i)$, I know that the standard absolute value on $\mathbb{C}$ is in $M_k$ but I can't imagine what absolute value satisfies that whose restriction to $\mathbb{Q}$ is $|\cdot|_p$ for a prime number $p$.

Does there exists? Could you give me any example?

$\endgroup$
  • $\begingroup$ I think that I found at least one example. $\endgroup$ – LWW Jul 14 '18 at 4:30
1
$\begingroup$

Taking your example $k=\Bbb Q(i)$, then there

  • is one absolute value $|\cdot|_{\tilde p}$ on $k$ extending $|\cdot|_p$ if $p \equiv 3$ mod $4$, or $p=2$: it is given explicitly by $$|a+bi|_{\tilde p} = \sqrt{|a^2+b^2|_p}$$
  • are two absolute values $|\cdot|_{\pi_1}$, $|\cdot|_{\pi_2}$ on $k$ extending $|\cdot|_p$ if $p \equiv 1$ mod 4. Namely, in this case, $(p) = I_1\cdot I_2$ as ideals where $I_i$ are two different prime ideals in $\Bbb Z[i]$; choosing as $\pi_i$ any generator of $I_i$, the $\pi$-adic value first on $\Bbb Z[i]$ and then on its quotient field $k$ is defined completely analogously to $p$-adic values on $\Bbb Q$. (As examples, for $p=5$ one can choose $\pi_1 = 2+i, \pi_2=2-i$; for $p=13$ e.g. $\pi_1=2+3i, \pi_2 = 2-3i$ for $p=13$).

All this is related (and from a certain perspective, nearly equivalent) to the decomposition/ramification of the prime $p$ in the number field $k$. For your specific example, compare e.g. What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$, Primes in Gaussian Integers, Classification of prime ideals in $\mathbb{Z}[i]$, Are there any elegant methods to classify of the Gaussian primes?.

$\endgroup$
  • $\begingroup$ Oh,I read other source so I know that example and in fact now I can describe all the absolute values but your answer is very nice and easy. I think that this helps many people who ask the same one I asked. Thank you. $\endgroup$ – LWW Jul 19 '18 at 1:33

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.