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Consider two algebraic integers $\alpha, \beta$,
and assume
\[
  [\Qbb[\alpha + \beta] : \Qbb] = [\Qbb(\alpha) :
  \Qbb] \times [\Qbb(\beta) : \Qbb]
.\]
This means that the Galois-conjugates are $\alpha_i
+ \beta_j$ where $\alpha_i$ runs through the
conjugates of $\alpha$ and $\beta_j$ runs through
the conjugates of $\beta$.

Then
\begin{align*}
  \mahler( \minp {\alpha + \beta})
  &= \prod_{i, j}
  \max(1, |\alpha_i + \beta|) \\
  &\le \prod_{i, j} 2
  \max(1, |\alpha_i|) \max(1, |\beta_j|) \\
  &= 2^{d_1 d_2} \left( \prod_{i}  \max(1,
  |\alpha_i|) \right)^{d_2} \left( \prod_{j}
  \max(1, |\beta_i|) \right)^{d_1} \\
  &= 2^{d_1 d_2} \mahler( \minp \alpha)^{d_2}
  \mahler(\minp \beta)^{d_1}
\end{align*}
Recall that we mentioned that we could define
\[
  H(\alpha) = \mahler( \minp \alpha)^{\frac{1}{\deg
  \minp \alpha}}
.\]
Then would have
\[
  H(\alpha + \beta) \le 2 H(\alpha)H(\beta)
.\]
Similarly,
\[
  H(\alpha \beta) \le H(\alpha) H(\beta)
.\]

\begin{proposition*}
  Let $P \in \Cbb[X]$ of degree $d$. Then
  \[
    2^{-d} \height(P) \le \mahler(P) \le (d + 1) \height(P)
  .\]
\end{proposition*}

\begin{proof}
  For the upper bound:
  \[
    \log M(P) = \int_0^1 \log |P(e^{-2\pi i t})|
    \dd t
  .\]
  Known as Jensen's formula (enough to prove for
  $P$ of degree $1$).

  Note that
  \[
    |P(X)| \le (d + 1)H(P)
  \]
  for all $|X| = 1$. This with Jensen's formula
  gives the upper bound. For the lower bound:
  \[
    P(X) = a_d X^d + \cdots + a_1 X + a_0
  .\]
  Then
  \[
    \left| \frac{a_j}{a_d} \right| = \sum_{\{k_1,
    \ldots, k_j\} \subset \{1, \ldots, d\}}
    \ub{|a_{k_1}| \cdots |a_{k_j}|}_{\le
    \mahler(f) / |a_d|}
  .\]
  The number of terms is $\le 2^d$. Hence $|a_j|
  \le 2^d \mahler(P)$.
\end{proof}

\subsubsection*{Absolute Values}

Let $K$ be a number field. Then a function
$|\bullet| : K \to \Rbb_{\ge 0}$ is an absolute
value if:
\begin{itemize}
  \item $|\alpha\beta| = |\alpha||\beta|$
  \item $|\alpha + \beta| \le |\alpha| + |\beta|$
    for all $\alpha, \beta \in K$
\end{itemize}

\begin{example*}
  \phantom{}
  \begin{itemize}
    \item Trivial absolute value: $|\alpha| = 0$
      for all $\alpha \in K$.
    \item Let $\sigma : K \to \Cbb$ be an
      embedding. Then $|\alpha|_\sigma =
      |\sigma(\alpha)|$.
    \item Let $P \subset \O_K$ be a non-zero prime
      ideal lying above $p \in \Zbb$. (This means
      $p \in P$).

      Then we define $\operatorname{ord}_P$ on $K$ as follows:
      for $\alpha \in \O_K$,
      $\operatorname{ord}_P(\alpha)$ is the
      largest $m$ such that $P^m \mid \alpha
      \O_K$. For $\alpha, \beta \in \O_K$,
      $\operatorname{ord}_P(\alpha / \beta) =
      \operatorname{ord}_P(\alpha) -
      \operatorname{ord}_p(\beta)$.

      Let $e_P = \operatorname{ord}_P(p)$
      (ramification index). Then we define
      \[
        |\alpha|_P =
        p^{-\operatorname{ord}_P(\alpha) / e_p}
      .\]
  \end{itemize}
\end{example*}

Comment on the normalisation: for $\alpha \in
\Qbb$, we have $|\alpha|_\sigma =
|\alpha|_\infty$, and $|\alpha|_P =
|\alpha|_p$.

The places of $K$ are $M_K$ comprises:
\begin{itemize}
  \item all embeddings $\sigma : K \to \Cbb$ such
    that $\sigma(K) \in \Rbb$
  \item one from each complex conjugate pairs from
    the rest
  \item all (non-zero) prime ideals
\end{itemize}

For $v \in M_K$, $|\bullet|_v$ denotes the
absolute value given above.

Infinite places: $M_{K, \infty}$: embeddings.

Finite places: $M_{K, f}$: prime ideals.

For $v \in M_K$, we define $d_v$ as follows:
\begin{itemize}
  \item if $v$ is a real embedding, then $d_v =
    1$.
  \item if $v$ is complex, then $d_v = 2$.
  \item if $v$ is a prime ideal, then $d_v = e_v
    \cdot f_v$, where: $[\O_K / v : \Zbb / p\Zbb]
    = f_v$ (where $p$ is the rational prime below
    $v$).
\end{itemize}

Comment:
\[
  d_v = [K : \Qbb_p]
\]
where $p$ is the place of $\Qbb$ below $v$.
