{
  "statement": "Prove H3 below. No checkpoint result is granted.\n\n**Target: H3**\n\nThere exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\n\nHere \\(\\operatorname{rd}(L)=|\\operatorname{Disc}(L)|^{1/[L:\\mathbb Q]}\\) is the root discriminant of a number field \\(L\\), and all logarithms are natural.",
  "starting_statement": null,
  "target_statement": "There exists an absolute constant \\(C\\ge1\\) such that, for every totally real number field \\(F\\) of degree \\(f=[F:\\mathbb Q]\\), the field\n\\[\nK=F(\\mathrm i),\\qquad \\mathrm i^2=-1,\n\\]\nsatisfies\n\\[\n\\operatorname{rd}(K)\\le2\\operatorname{rd}(F),\n\\qquad\nh(K)\\le\\bigl(C\\operatorname{rd}(F)\\bigr)^{2f}.\n\\]\n\nHere \\(h(K)\\) is the ideal class number of the ring of integers of \\(K\\). The constant \\(C\\) is independent of \\(F\\) and \\(f\\).\n\nHere \\(\\operatorname{rd}(L)=|\\operatorname{Disc}(L)|^{1/[L:\\mathbb Q]}\\) is the root discriminant of a number field \\(L\\), and all logarithms are natural.",
  "case_id": "H0-to-H3",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}