How can I draw the following diagram from field theory?

Question

How can I draw the following diagram using TikZ/PGF?

I have been able to draw up to the following diagram:

Using the following code:

\[
  \begin{tikzpicture}[node distance = 2cm, auto]
  \node (Q) {$\mathbb{Q}$};
  \node (E) [above of=Q, left of=Q] {$\mathbb{Q}(\sqrt{2})$};
  \node (E1) [above of=Q] {$\mathbb{Q}(\sqrt{6})$};
  \node (F) [above of=Q, right of=Q] {$\mathbb{Q}(\sqrt{3})$};
  \node (K) [above of=Q, node distance = 4cm] {$\mathbb{Q}(\sqrt{2}, \sqrt{3})$};
  \draw[-] (Q) to node {$2$} (E);
  \draw[-] (Q) to node [swap] {$2$} (F);
  \draw[-] (E) to node {$2$} (K);
  \draw[-] (F) to node [swap] {$2$} (K);
  \draw[-] (Q) to node [swap] {$2$} (E1);
  \draw[-] (E1) to node [swap] {$2$} (K);
  \end{tikzpicture}
\]

How can I extend this code to the above one?

Answer 1

Using the code you already have, you could simply do something like this (but since the code you provided does not exactly result in the image you posted, I don't know what you really want to achieve):

\documentclass[tikz, border=1mm]{standalone}
\usepackage{amssymb}

\begin{document}
 
\begin{tikzpicture}[node distance = 2cm, auto]
    \node (Q) {$\mathbb{Q}$};
    \node (E) [above of=Q, left of=Q] {$\mathbb{Q}(\sqrt{2})$};
    \node (E1) [above of=Q] {$\mathbb{Q}(\sqrt{6})$};
    \node (F) [above of=Q, right of=Q] {$\mathbb{Q}(\sqrt{3})$};
    \node (K) [above of=Q, node distance = 4cm] {$\mathbb{Q}(\sqrt{2}, \sqrt{3})$};

    \node (G) [right of=F] {\color{blue}$\mathbb{Q}(\sqrt{\gamma})$};

    \draw[-] (Q) to node {$2$} (E);
    \draw[-] (Q) to node {$2$} (F);
    \draw[-] (E) to node {$2$} (K);
    \draw[-] (F) to node {$2$} (K);
    \draw[-] (Q) to node {$2$} (E1);
    \draw[-] (E1) to node {$2$} (K);

    \draw[-] (Q) to node [swap] {$3$} (G);
    \draw[-] (G) to node [swap] {$2$} (K);
\end{tikzpicture}

\end{document}

---

Edit

In case you need the vertical offset of the nodes as well, you could achieve this by using a matrix, or (as noted in the comments) using tikz-cd:

\documentclass[tikz, border=1mm]{standalone}
\usetikzlibrary{matrix}
\usepackage{amssymb}

\begin{document}
 
\begin{tikzpicture}

    \matrix (m) [matrix of math nodes, row sep=2em, column sep=1em] {
      & \color{blue}\mathbb{Q}(\zeta, \sqrt[3]{2}) & & \\
      & \mathbb{Q}(\sqrt[3]{2}) & \mathbb{Q}(\zeta\sqrt[3]{2}) & \mathbb{Q}(\zeta^2\sqrt[3]{2}) \\
      \color{blue}\mathbb{Q}(\zeta) & & & \\
      & \color{blue}\mathbb{Q} & & \\
    };

    \begin{scope}[every node/.style={font=\footnotesize}]
        \draw (m-1-2) to node[left] {3} (m-3-1)
                      to node[below left] {2} (m-4-2);
        \draw (m-1-2) to node[right] {2} (m-2-2)
                      to node[right] {3} (m-4-2);
        \draw (m-1-2) to node[below left] {2} (m-2-3)
                      to node[right] {3} (m-4-2);
        \draw (m-1-2) to node[above right] {2} (m-2-4)
                      to node[below right] {3} (m-4-2);
    \end{scope}

\end{tikzpicture}

\end{document}

Answer 2

Instead of extending your code, I use plain TikZ. We even can write the one-liner code for the figure ^^

\documentclass[tikz,border=5mm]{standalone}
\usepackage{amsmath,amssymb}
\begin{document}
\begin{tikzpicture}
\path
(0,0) node (C) {$\mathbb{Q}(\sqrt[3]{2})$}
(2,0) node (R1) {$\mathbb{Q}(\zeta\sqrt[3]{2})$}
(4,0) node (R2) {$\mathbb{Q}(\zeta^2\sqrt[3]{2})$}
(0,1.5) node[blue] (A) {$\mathbb{Q}(\zeta,\sqrt[3]{2})$}
(0,-2) node[blue] (B) {$\mathbb{Q}$}
(-2,-1)  node[blue] (L) {$\mathbb{Q}(\zeta)$}
;
\draw[nodes={scale=.8}] (A)
to node[above]{$2$} (R2)
to node[below]{$3$} (B)
to node[below]{$2$} (L)
to node[left]{$3$} (A)
to node[below]{$2$} (R1)
to node[above]{$3$} (B)
to node[right]{$3$} (C)
to node[right]{$2$} (A)
;       
\end{tikzpicture}
\end{document}

Answer 3

For completeness, here is one made using tikz-cd

\documentclass[tikz,border=5mm]{standalone}
\usepackage{amsmath,amssymb,tikz-cd}
\usetikzlibrary{babel} % incase " is active
\begin{document}
\begin{tikzcd}
  &
  \mathbb{Q}(\sqrt[3]{2})
  \arrow[ddl,-,"3"']
  \arrow[d,-,"2"]
  \arrow[dr,-,"2"']
  \arrow[drr,-,"2"]
  &
  &
  \\
  &
  \mathbb{Q}(\sqrt[3]{2})
  \arrow[dd,-,"3"]
  &
  \mathbb{Q}(\zeta,\sqrt[3]{2})
  \arrow[ddl,-,"3"']
  &
  \mathbb{Q}(\zeta^2\sqrt[3]{2})
  \arrow[ddll,-,"3"]
  \\
  \mathbb{Q}(\zeta)
  \arrow[dr,-,"2"']
  &
  &
  &
  \\
  &
  \mathbb{Q}
  &
  &
  \\
\end{tikzcd}
\end{document}