Let $K$ be a <number field>. Its <place of a number field>[places] consist of its real embeddings, conjugate pairs of complex embeddings, and the finite places associated with nonzero <prime ideals> $\mathfrak p$ of its <ring of integers of a number field>. At a real or complex place use the usual absolute value. If $\mathfrak p$ lies over the <prime number> $p$ with <ramification index> $e_{\mathfrak p}$, normalize its <non-Archimedean place of a number field>[absolute value] by
$$
|x|_{\mathfrak p}=p^{-\operatorname{ord}_{\mathfrak p}(x)/e_{\mathfrak p}}.
$$
These normalizations extend the standard absolute values on $\mathbb Q$.

Write $d_v=[K_v:\mathbb Q_v]$ for the <local degree of a place>. Thus $d_v$ is $1$ at a real place, $2$ at a complex place, and $e_vf_v$ at a finite place. The <Absolute multiplicative Weil height> is
$$
H(\alpha)=
\prod_{v\in M_K}\max\{1,|\alpha|_v\}^{d_v/[K:\mathbb Q]}.
$$
The <product formula> shows that this is unchanged when $K$ is replaced by a larger number field containing $\alpha$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 166 / 2 / a / Solution

If a prime ideal $\mathfrak p$ of $K$ lies over the <prime number> $p$ with <ramification index> $e_{\mathfrak p}$, its normalized absolute value is
$$
|x|_{\mathfrak p}=p^{-\operatorname{ord}_{\mathfrak p}(x)/e_{\mathfrak p}}.
$$
It 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>.

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 Incoming links: Non-Archimedean place of a number field