So far answers focused on several ways to draw your diagrams with tikz
. This answer will focus on a more holistic approach to create and maintain your document to address the question shown, but not verbally asked. Again, this is just one way to do it.
This is what we want to achieve in the end:

1) Have 1 file for each tikz-drawing
Pick any of the previous solutions, e.g. mine. Create and run/compile circles.tex
:
\documentclass[10pt,border=3mm,tikz]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}% for the arrows in first diagram
\usepackage{amssymb}
\begin{document}
\begin{tikzpicture}[
execute at begin node=$,
execute at end node=$,% this way you can omit the $..$
> = {stealth}]
% ~~~ circles ~~~~~~~~~~~~~~
\draw (0,0) circle(1.25);
\draw (0,0) circle(2.75);
\draw (0,0) circle(4.25);
% ~~~ Text / nodes ~~~~~~
\node at (0,0) {F = \mathbb{Q}};
\node at (0,1.8) {E = \mathbb{Q}(\sqrt{5})};
\node at (0,3.5) {K = \mathbb{Q}(\sqrt{5},i)};
% ~~~+ arcs ~~~~~~~~~~~~~~~~
% [->] draws a line with an arrow, while [gray] and [dotted] indicate ideas for you
\draw [->,gray] (-45:1.8) arc[start angle=-45,end angle=0,radius=1.8];
\draw [->,dotted] (-45:3.5) arc[start angle=-45,end angle=0,radius=3.5];
\end{tikzpicture}
\end{document}

Create and run map.tex
:
\documentclass[10pt,border=3mm,tikz]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}% for the arrows
\usetikzlibrary{positioning}% for the mapping diagram
%\usepackage{amssymb}% not needed here
\begin{document}
\begin{tikzpicture}[> = {Stealth}]
% ~~~ nodes ~~~~
\node (K) {K};% node's name is K, to be referenced later
\node[below=of K] (E) {E};
\node[below=of E] (F) {F};
\node[right=of K] (GK) {Gal(K/K)};
\node[right=of E] (GE) {Gal(K/E)};
\node[right=of F] (GF) {Gal(K/F)};
% ~~~ lines ~~~~~~~~~~~
\draw[dashed] (K) -- (GK);% () means: recall positions of nodes named K, GK etc.
\draw[dashed] (E) -- (GE);
\draw[dashed] (F) -- (GF);
\draw[->] (F) -- (E);
\draw[->] (E) -- (K);
\draw[->] (GK) -- (GE);
\draw[->] (GE) -- (GF);
\end{tikzpicture}
\end{document}

A few remarks:
- you'll have
circles.pdf
and map.pdf
- kindly notice the standalone class
- kindly notice the variations in packages used, as needed per drawing
2) Include these .pdf's in your main document
Create your main document, say mathDoc.tex
as given below. Once you compile it, you'll have the intended result. Remarks after the code.
\documentclass[10pt,a4paper]{article}
\usepackage{graphicx}% to include the tikz-drawings as pdf
\usepackage{subfigure}% to have subfigures
\usepackage{lipsum}% blindtext
\usepackage{varioref}% nice referencing; requires 2 compile cycles, too
% ~~~ some shorthand notation ~~~~
\newcommand\myfig[1]{figure \vref{fig:#1}}% referencing figures
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\begin{document}
\lipsum[1]% some blindtext
See \myfig{demo} for more details.
\begin{figure}[h]% tries to put figure "here"
\centering
\subfigure[Circles]{\includegraphics[width=.65\textwidth]{circles}}
\subfigure[Map]{\includegraphics[width=.25\textwidth]{map}}
\caption{The intended diagrams\label{fig:demo}}
\end{figure}
\lipsum[2]% some blindtext
\end{document}
Kindly notice:
- the different documentclass used
- the packages used, only relevant for mathDoc.tex
- the 2 subfigures
- including the pdf's from above tikz.drawings
- adaptation of graphics-width via
\includegraphics[width=.65\textwidth]{..}
Impact on your workflow
Now you can create and adjust tikz-drawings and have them decoupled from your main document. I.e. edit and compile each tikz-diagram as long and often as you need. Finally run the main document to include the results.
Vice versa you can focus on the flow of content in your main document first and create missing tikz drawings later.
The layout burden
As you can see you encounter what every graphic designer has to deal with when doing the layout, say for a newspaper. Space is limited. So wrt drawings they'll specify at least available space: "fit your foto or drawing within x-y cm^2, while I don't care how you do it: just have it readable".
So you may need to work a bit back and forth between the diagrams width-specification and their tikz-code, to have good readability.
Improving organization
To avoid too much clutter of your main directory, you may want to work with folders, say /tikz
which will provide all your tikz-drawing.
All you then need to do is telling Latex where to find your includes, like so:
\includegraphics[width=.65\textwidth]{tikz/circles}
circuitikz
environment, just atikzpicture
is already good. The packagecircuitikz
is built upon TikZ. You should use\tikzset{}
and nottikzstyle
, so replace\tikzstyle{every node}=[font=\normalsize]
with\tikzset{every node/.append style={font=\normalsize}}
. If you want to scale the whole thing, add the optionscale
to the\tikzset
command and add a value lower than 1. Finally, for the curved arrows, I would suggest that you use anarc
and polar coordinates.