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}

tikz-cd
as it is made for this kind of stuff.tikz-cd
tikz-cd
installed you can use it no matter which editor. And if doing diagrams like this you should use it.tikz-cd
is very helpful for mathematical diagram. I tried in the past but failed to install it. Can you give some precise idea how to addtikz-cd
with my existing latex editor ?