21_signed_norm.tex

Paper X: Universal Residual Seeds and Conditional Occupied-System Completion

21_signed_norm.tex

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \draw[black!13,line width=.25pt] (1.5,2.75) -- (1.5,2.76);
4 \draw[] (1.5,2.69) -- (1.5,2.81);
5 \node[font=\small,align=center,font=\scriptsize] at (1.5,2.49) {-5};
6 \draw[black!13,line width=.25pt] (3.409091,2.75) -- (3.409091,2.76);
7 \draw[] (3.409091,2.69) -- (3.409091,2.81);
8 \node[font=\small,align=center,font=\scriptsize] at (3.409091,2.49) {-3};
9 \draw[black!13,line width=.25pt] (6.272727,2.75) -- (6.272727,2.76);
10 \draw[] (6.272727,2.69) -- (6.272727,2.81);
11 \node[font=\small,align=center,font=\scriptsize] at (6.272727,2.49) {0};
12 \draw[black!13,line width=.25pt] (8.181818,2.75) -- (8.181818,2.76);
13 \draw[] (8.181818,2.69) -- (8.181818,2.81);
14 \node[font=\small,align=center,font=\scriptsize] at (8.181818,2.49) {2};
15 \draw[black!13,line width=.25pt] (11.045455,2.75) -- (11.045455,2.76);
16 \draw[] (11.045455,2.69) -- (11.045455,2.81);
17 \node[font=\small,align=center,font=\scriptsize] at (11.045455,2.49) {5};
18 \draw[->,line width=.6pt] (1.5,2.75) -- (12.12,2.75);
19 \draw[->,line width=.6pt] (1.5,2.75) -- (1.5,2.88);
20 \node[font=\small,align=center,font=\scriptsize] at (6.75,2.2) {integer generators};
21 \path[draw,fill=black!15] (3.409091,2.75) circle (0.12);
22 \draw[->] (3.409091,2.55) -- (6.55,1.75);
23 \path[draw,fill=black!15] (8.181818,2.75) circle (0.12);
24 \draw[->] (8.181818,2.55) -- (6.55,1.75);
25 \node[font=\small,align=center] at (6.55,1.45) {same residue $2\pmod5$};
26 \node[font=\small,align=center] at (3,4.35) {$N_{\mathbb Q/\mathbb Q}(-3)=-3$};
27 \node[font=\small,align=center] at (10.5,4.35) {$N_{\mathbb Q/\mathbb Q}(2)=2$};
28 \node[font=\small,align=center] at (3,0.45) {$N((-3))=3$};
29 \node[font=\small,align=center] at (10.5,0.45) {$N((2))=2$};
30 \node[font=\small,align=center,font=\footnotesize] at (7,-0.15) {The ideal norm is positive. Passing to absolute values must retain the real sign.};
31 \end{tikzpicture}

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