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For coprime integers $p,q$ with $q>0$,This follows directly from the real and p-adic absolute values, or from the height-Mahler measure formula for the primitive polynomial $qX-p$.

If a prime ideal $\mathfrak{p}$ of $K$ lies over the prime number $p$ with ramification index $e_{\mathfrak{p}}$, its normalized absolute value isIt extends the usual p-adic absolute value on $\mathbb{Q}$. Its local degree is $[K_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}{f}_{\mathfrak{p}}$, where $f_{\mathfrak{p}}$ is the residue-field degree.

For coprime integers $p,q$ with $q>0$,This follows either directly from the real and p-adic absolute values, or from the height-Mahler measure formula applied to the primitive minimal polynomial $qX-p$.

Solved by gpt-5.6-sol high.

Two absolute values are equivalent when $|\cdot {|}_2=|\cdot {|}_1^c$ for some $c>0$; equivalently, they induce the same topology. The nontrivial non-Archimedean absolute values on $\mathbb{Q}$ are, up to equivalence, exactly the p-adic absolute value $|\cdot {|}_p$.

Indeed $|n|\le 1$ for every integer $n$. Nontriviality gives a prime $p$ with $|p|<1$. If $(m,p)=1$, choose $a_r,b_r\in \mathbb{Z}$ with $a_{r}m+b_r{p}^r=1$. For large $r$, $|b_r{p}^r|<1$, so the ultrametric inequality forces $|m|=1$. HenceIf no prime has absolute value below one, the absolute value is trivial. This proves the non-Archimedean part of Ostrowski theorem.

Solved by gpt-5.6-sol high.