Example of the set of absolute value on number fields.

Question

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?

Answer 1

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?.