{
  "statement": "You may assume H2 below without proof. Prove H3 below. No other checkpoint result is granted.\n\n**Given: H2**\n\nThere exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), putting\n\\[\nt=\\left\\lfloor\\frac{(\\ell-1)^2}{100}\\right\\rfloor,\n\\]\nthere exist a totally real cyclic cubic field \\(F/\\mathbb Q\\), distinct rational primes \\(q_1,\\ldots,q_t\\), and a tower\n\\[\nF=F_0\\subset F_1\\subset F_2\\subset\\cdots\n\\]\nsatisfying all the following properties.\n\n1. Every \\(q_b\\) is congruent to \\(1\\pmod4\\) and splits completely in every \\(F_j\\), for every \\(j\\ge0\\).\n\n2. Every \\(F_j/F\\) is finite Galois and everywhere unramified, and its Galois group is a \\(3\\)-group. Every \\(F_j\\) is totally real.\n\n3. Writing \\(f_j=[F_j:\\mathbb Q]\\), one has\n\\[\nf_j\\longrightarrow\\infty,\n\\qquad\n\\operatorname{rd}(F_j)=\\operatorname{rd}(F),\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nFor each fixed \\(\\ell\\), the primes \\(q_1,\\ldots,q_t\\) are chosen once and are independent of \\(j\\).\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.\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": "There exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), putting\n\\[\nt=\\left\\lfloor\\frac{(\\ell-1)^2}{100}\\right\\rfloor,\n\\]\nthere exist a totally real cyclic cubic field \\(F/\\mathbb Q\\), distinct rational primes \\(q_1,\\ldots,q_t\\), and a tower\n\\[\nF=F_0\\subset F_1\\subset F_2\\subset\\cdots\n\\]\nsatisfying all the following properties.\n\n1. Every \\(q_b\\) is congruent to \\(1\\pmod4\\) and splits completely in every \\(F_j\\), for every \\(j\\ge0\\).\n\n2. Every \\(F_j/F\\) is finite Galois and everywhere unramified, and its Galois group is a \\(3\\)-group. Every \\(F_j\\) is totally real.\n\n3. Writing \\(f_j=[F_j:\\mathbb Q]\\), one has\n\\[\nf_j\\longrightarrow\\infty,\n\\qquad\n\\operatorname{rd}(F_j)=\\operatorname{rd}(F),\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nFor each fixed \\(\\ell\\), the primes \\(q_1,\\ldots,q_t\\) are chosen once and are independent of \\(j\\).\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.",
  "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": "H2-to-H3",
  "given_context": {
    "premise": "There exist an absolute constant \\(C>0\\) and an integer \\(\\ell_0\\ge2\\) such that, for every integer \\(\\ell\\ge\\ell_0\\), putting\n\\[\nt=\\left\\lfloor\\frac{(\\ell-1)^2}{100}\\right\\rfloor,\n\\]\nthere exist a totally real cyclic cubic field \\(F/\\mathbb Q\\), distinct rational primes \\(q_1,\\ldots,q_t\\), and a tower\n\\[\nF=F_0\\subset F_1\\subset F_2\\subset\\cdots\n\\]\nsatisfying all the following properties.\n\n1. Every \\(q_b\\) is congruent to \\(1\\pmod4\\) and splits completely in every \\(F_j\\), for every \\(j\\ge0\\).\n\n2. Every \\(F_j/F\\) is finite Galois and everywhere unramified, and its Galois group is a \\(3\\)-group. Every \\(F_j\\) is totally real.\n\n3. Writing \\(f_j=[F_j:\\mathbb Q]\\), one has\n\\[\nf_j\\longrightarrow\\infty,\n\\qquad\n\\operatorname{rd}(F_j)=\\operatorname{rd}(F),\n\\qquad\n\\log\\operatorname{rd}(F)\\le C\\ell\\log\\ell.\n\\]\n\nFor each fixed \\(\\ell\\), the primes \\(q_1,\\ldots,q_t\\) are chosen once and are independent of \\(j\\).\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."
  },
  "protocol": "single"
}