% This picture represents the lattice antisomorphism between the lattice of the 
% subgroups of the Galois group of a Galois extension and the lattice of
% intermediary subfields of such extension.
% Copyright Pablo (C) 2023
\begin{tikzpicture}[ampersand replacement=\&]
  % The elements of both lattices
  \matrix(m)[matrix of math nodes,row sep=2em,column sep=2em,minimum width=2em]
  {     
    \& K \& \& 1 \& \\
    \& L_1 \cdot L_2 \& \& H_1 \cap H_2 \& \\
    L_1 \& \& H_1 \& \& \\
    \& \& L_2 \& \& H_2 \\
    \& L_1 \cap L_2 \& \& \langle H_1, H_2 \rangle \& \\
    \& k \& \& \operatorname{Gal}(K/k) \& \\
  };

    % The relations of both lattices
    \draw (m-1-2) -- (m-2-2)
                  -- (m-3-1)
                  -- (m-5-2)
                  -- (m-6-2)
          (m-2-2) -- (m-4-3)
                  -- (m-5-2);
    \draw (m-1-4) -- (m-2-4)
                  -- (m-3-3)
                  -- (m-5-4)
                  -- (m-6-4)
          (m-2-4) -- (m-4-5)
                  -- (m-5-4);

    % The antisomorphism
    \draw[dotted] (m-1-2) -- (m-1-4)
                  (m-2-2) -- (m-2-4) 
                  (m-3-1) -- (m-3-3)
                  (m-4-3) -- (m-4-5)
                  (m-5-2) -- (m-5-4)
                  (m-6-2) -- (m-6-4);
\end{tikzpicture}

