Galois correspondence — splitting field of x⁴ − 2 over ℚ

/* Galois correspondence for the splitting field of x^4 - 2 over Q.
   K = Q(2^(1/4), i),  [K:Q] = 8,  Gal(K/Q) = D4 = <r,s | r^4 = s^2 = 1, srs = r^-1>
   r : 2^(1/4) -> i*2^(1/4), i -> i        s : complex conjugation
   The correspondence reverses inclusion: |H| * [Fix(H):Q] = 8. */
graph galois_x4_minus_2 {
  graph [bgcolor=transparent, newrank=true, ranksep=0.6, nodesep=0.4,
         fontname="Inter", fontsize=11,
         label="Galois correspondence for K = Q(2^(1/4), i) over Q — Gal(K/Q) = D4, order 8\nBlue = normal subgroup / Galois subextension.  Amber = non-normal subgroup / non-Galois subfield.",
         labelloc=b];
  node [shape=box, style="rounded,filled", fontname="Inter", fontsize=10, penwidth=0, margin="0.12,0.06"];
  edge [color="#94a3b8", penwidth=1.1];

  subgraph cluster_groups {
    label="Subgroups of D4"; fontname="Inter"; fontsize=12; color="#cbd5e1";
    node [fillcolor="#dbeafe"];
    gD4  [label="D4 = <r, s>\norder 8"];
    gR   [label="<r> = Z/4Z\norder 4"];
    gV1  [label="<r^2, s> = V4\norder 4"];
    gV2  [label="<r^2, rs> = V4\norder 4"];
    gR2  [label="<r^2>\norder 2"];
    gE   [label="{1}\norder 1"];
    gS   [label="<s>\norder 2", fillcolor="#fef3c7"];
    gR2S [label="<r^2 s>\norder 2", fillcolor="#fef3c7"];
    gRS  [label="<r s>\norder 2", fillcolor="#fef3c7"];
    gR3S [label="<r^3 s>\norder 2", fillcolor="#fef3c7"];

    gD4 -- gR; gD4 -- gV1; gD4 -- gV2;
    gR  -- gR2;
    gV1 -- gR2; gV1 -- gS;  gV1 -- gR2S;
    gV2 -- gR2; gV2 -- gRS; gV2 -- gR3S;
    gR2 -- gE; gS -- gE; gR2S -- gE; gRS -- gE; gR3S -- gE;
  }

  subgraph cluster_fields {
    label="Intermediate fields Q <= F <= K"; fontname="Inter"; fontsize=12; color="#cbd5e1";
    node [fillcolor="#dbeafe"];
    fQ    [label="Q\ndegree 1"];
    fQi   [label="Q(i)\ndegree 2"];
    fQ2   [label="Q(sqrt 2)\ndegree 2"];
    fQm2  [label="Q(sqrt -2)\ndegree 2"];
    fQ2i  [label="Q(sqrt 2, i)\ndegree 4"];
    fK    [label="K = Q(2^(1/4), i)\ndegree 8"];
    fQ4   [label="Q(2^(1/4))\ndegree 4", fillcolor="#fef3c7"];
    fQi4  [label="Q(i 2^(1/4))\ndegree 4", fillcolor="#fef3c7"];
    fQp   [label="Q((1+i) 2^(1/4))\ndegree 4", fillcolor="#fef3c7"];
    fQm   [label="Q((1-i) 2^(1/4))\ndegree 4", fillcolor="#fef3c7"];

    fQ   -- fQi; fQ -- fQ2; fQ -- fQm2;
    fQi  -- fQ2i;
    fQ2  -- fQ2i; fQ2  -- fQ4;  fQ2  -- fQi4;
    fQm2 -- fQ2i; fQm2 -- fQp;  fQm2 -- fQm;
    fQ2i -- fK; fQ4 -- fK; fQi4 -- fK; fQp -- fK; fQm -- fK;
  }

  { rank=same; gD4; fQ; }
  { rank=same; gR; gV1; gV2; fQi; fQ2; fQm2; }
  { rank=same; gR2; gS; gR2S; gRS; gR3S; fQ2i; fQ4; fQi4; fQp; fQm; }
  { rank=same; gE; fK; }
}