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  For a polynomial $P \in K[X]$, we write $|P|_v$
  for the maximum $|\bullet|_v$ of all the
  coefficients of $P$.

  A variant of Gauss's lemma can be stated as follows: Let
  $Q_1, Q_2 \in K[X]$. Then $|Q_1 Q_2|_v = |Q_1|_v
  |Q_2|_v$ for $v \in M_{K, f}$.

  Observe that $|f_\alpha|_v = 1$ (for all $v \in
  M_{K, f}$) because the
  coefficients are coprime rational integers. We
  write $f_\alpha = a_d (X - \alpha_1) \cdots (X -
  \alpha_d)$ (we take $K$ to be the splitting
  field of $f_\alpha$). Gauss's lemma gives
  \[
    \prod_{v \in M_{K, f}} |a_d|_v^{d_v} \cdot \prod_{v
    \in K_{K, f}} \prod_{j = 1}^{d} \max(1,
    |\alpha_j|_v))^{d_v} = 1
  .\]
  Let $\sigma$ be an automorphism of $K$ such that
  $\sigma \alpha_j = \alpha$ for some fixed $j$.
  This permutes $M_{K, f}$. That is, $\forall v
  \in M_{K, f}$, there exists $\sigma v \in M_{K,
  f}$ such that $|\sigma \beta|_{\sigma v} =
  |\beta|_v$. So
  \begin{align*}
    \prod_{v \in M_{K, f}} \max(1,
    |\alpha_j|_v)^{d_v}
    &= \prod_{v \in M_{K, f}}
    \max(1, |\ub{\sigma
    \alpha_j}_{=\alpha}|_{\sigma v})^{d_{\sigma
    v}} \\
    &= \prod_{v \in M_{K, f}} \max(1,
    |\alpha|_v)^{d_v}
  \end{align*}
  By the product formula:
  \[
    \prod_{v \in M_{K, f}} |a_d|_v^{d_v} = \prod_{v
    \in M_{K, \infty}} |a_d|_v^{-d_v} = |a_d|^{-[K
    : \Qbb]}
  .\]
  So
  \[
    \left[ \prod_{v \in M_{K, f}} \max(1,
    |\alpha|_v)^{d_v}
    \right]^{\ob{d}^{=[\Qbb(\alpha) : \Qbb]}}
    = |a_d|^{[K : \Qbb]}
    \qedhere
  .\]
\end{proof}

\begin{lemma*}
  Let $\alpha \in \ol{\Qbb}$, and $k \in \Zbb$.
  Then
  \[
    H(a^k) = H(\alpha)^{|k|}
  .\]
\end{lemma*}

\begin{proof}
  If $k > 0$, then this is immediate from the
  definition. So just need to consider $k = -1$:
  \[
    H(\alpha^{-1})^d = \prod_{v \in M_K} \max(1,
    |\alpha|_v^{-1})^{d_v}
  \]
  ($d = \deg \alpha$). We multiply this by
  \[
    \prod_{v \in M_v} |\alpha|_v^{d_v} = 1
  .\]
  So
  \[
    H(\alpha^{-1})^d = \prod_{v \in M_k}
    \max(|\alpha|_v, 1)^{d_v} = H(\alpha)^d
  .\]
\end{proof}

Let $P$ be a polynomial in possibly several
variables, with complex coefficients. Then
$\mathcal{L}(P)$ is defined to be the sum of the
absolute values of all the coefficients. This is
sometimes called the length of $P$.

\begin{proposition*}
  Let $k \in \Zbb_{> 1}$, $n_1, \ldots, n_k \in
  \Zbb_{> 0}$. Let $P, Q \in \Zbb[X_1, \ldots,
  X_k]$ of degree $\le n_j$ in $X_j$. Let
  $\alpha_1, \ldots, \alpha_k \in \ol{\Qbb}_{\neq
  0}$. Then:
  \[
    H \left( \frac{P(\alpha_1, \ldots,
    \alpha_k)}{Q(\alpha_1, \ldots, \alpha_k)}
    \right) \le \max(\mathcal{L}(P), \mathcal{L}(Q))
    \cdot \prod_{j = 1}^{k} H(\alpha_j)^{k_j}
  .\]
  In particular: $H(\alpha\beta) \le
  H(\alpha)H(\beta)$ and $H(\alpha + \beta) \le
  2H(\alpha)H(\beta)$.
\end{proposition*}

\begin{proof}
  Let $K$ be a number field containing all
  $\alpha_i$.
  \begin{align*}
    H \left( \frac{P(\ldots)}{Q(\ldots)}
    \right)^{[K : \Qbb]}
    &= \prod_{v \in M_K} \max
    \left( 1, \left| \frac{P(\ldots)}{Q(\ldots)}
    \right|_v \right)^{d_v} \\
    &= \prod_{v \in M_K} \max(|Q(\ldots)|_v,
    |P(\ldots)|_v)^{d_v}
    &&\text{from product formula for $Q(\ldots)$}
  \end{align*}
  Let first $v \in M_{K, f}$. Then
  \begin{align*}
    |P(\alpha_1, \ldots, \alpha_k)|_v
    &\le \max_{\substack{
      j_1 = 0, \ldots, n_1 \\
      \vdots \\
      j_k = 0, \ldots, n_k
    }}|\alpha_1|_v^{j_1} \cdots |\alpha_k|^{j_k}
    \\
    &= \prod_{i = 1}^{k} \max(1, |\alpha_i|_v^{n_i})
  \end{align*}
  For $v \in M_{K, \infty}$:
  \[
    |P(\alpha_1, \ldots, \alpha_k)|_v \le \mathcal{L}(P) \cdot \prod_{i =
    1}^{k} \max(1, |\alpha_i|_v^{n_i})
  .\]
  So
  \[
    H \left( \frac{P(\ldots)}{Q(\ldots)}
    \right)^{[K : \Qbb]} \le \max(\mathcal{L}(P),
    \mathcal{L}(Q))^{[K : \Qbb]}
    \prod_{i = 1}^{k}
    \prod_{v \in M_K}
    \max(1, |\alpha_i|_v)^{n_i d_v}
  .\]
  Then taking a $[K : \Qbb]$ root of both sides
  gives the desired inequality.
\end{proof}

\begin{lemma*}
  Let $\alpha \in \ol{\Qbb} \subset \Cbb$. Then:
  \[
    H(\alpha)^{-\deg \alpha} \le |\alpha| \le
    H(\alpha)^{\deg\alpha}
  .\]
\end{lemma*}

This is sometimes known as ``trivial bound'' or
``Liouville's bound''.

\begin{proof}
  \[
    H(\alpha)^{\deg \alpha} = \prod_{v \in M_K}
    \max(1, |\alpha|_v)^{d_v} \ge |\alpha|
  \]
  Apply this for $\alpha^{-1}$:
  \begin{align*}
    |\alpha^{-1}|
    &\le H(\alpha^{-1})^d = H(\alpha)^d \\
    |\alpha|
    &\ge H(\alpha)^{-d}
  \end{align*}
\end{proof}

\begin{theorem*}[Siegel]
  Let $\alpha$ be a real algebraic irrational
  number. Then for all $\eps > 0$, there exists $c
  = c(\alpha, \eps) > 0$ such that
  \[
    \left| \alpha - \frac{p}{q} \right| \ge
    cq^{-\sqrt{2d} - \eps}
  \]
  for all $p, q \neq 0 \in \Zbb$.
\end{theorem*}

We will spend the next 3-5 lectures proving this.
