 &= \tau^{-1}(k - \sqrt{2}) \\
    &= k + l \sqrt{2}.
  \end{align*}

  Similarly, for all \( k + l \sqrt{2} \in M \), we get

  \begin{align*}
    (\tau^{-1} \phi_3 \tau)(k + l \sqrt{2})
    &= (\tau^{-1} \phi_3)(k - \sqrt{2}) \\
    &= \tau^{-1}(k - \sqrt{2}) \\
    &= k + l \sqrt{2}.
  \end{align*}

  Here both \( \phi_1 \) and \( \phi_3 \) fix \( k - l \sqrt{2} \in
  \tau(M) \) because \( \phi_1, \phi_2 \in \tau(M)^* \), the set of \(
  \tau(M) \)-automorphisms of \( L \).
</p>
<p>
  Since both \( \tau^