latex-pictures
A collection of TikZ drawings and other images 🖌️
galois-lattice-antisomorphism.tikz (1281B)
1 % This picture represents the lattice antisomorphism between the lattice of the 2 % subgroups of the Galois group of a Galois extension and the lattice of 3 % intermediary subfields of such extension. 4 % Copyright Pablo (C) 2023 5 \begin{tikzpicture}[ampersand replacement=\&] 6 % The elements of both lattices 7 \matrix(m)[matrix of math nodes,row sep=2em,column sep=2em,minimum width=2em] 8 { 9 \& K \& \& 1 \& \\ 10 \& L_1 \cdot L_2 \& \& H_1 \cap H_2 \& \\ 11 L_1 \& \& H_1 \& \& \\ 12 \& \& L_2 \& \& H_2 \\ 13 \& L_1 \cap L_2 \& \& \langle H_1, H_2 \rangle \& \\ 14 \& k \& \& \operatorname{Gal}(K/k) \& \\ 15 }; 16 17 % The relations of both lattices 18 \draw (m-1-2) -- (m-2-2) 19 -- (m-3-1) 20 -- (m-5-2) 21 -- (m-6-2) 22 (m-2-2) -- (m-4-3) 23 -- (m-5-2); 24 \draw (m-1-4) -- (m-2-4) 25 -- (m-3-3) 26 -- (m-5-4) 27 -- (m-6-4) 28 (m-2-4) -- (m-4-5) 29 -- (m-5-4); 30 31 % The antisomorphism 32 \draw[dotted] (m-1-2) -- (m-1-4) 33 (m-2-2) -- (m-2-4) 34 (m-3-1) -- (m-3-3) 35 (m-4-3) -- (m-4-5) 36 (m-5-2) -- (m-5-4) 37 (m-6-2) -- (m-6-4); 38 \end{tikzpicture} 39