Cyclotomic Fields in Finite Geometry 2025, 34 - Modulus Equation

A CM-field is a totally imaginary quadratic extension of a totally real number field, and all cyclotomic fields are CM-fields. For a CM-field K, K = K⁺(√α), for K⁺ totally real, α ∈ K⁺ totally negative. It follows that complex conjugation induces an automorphism of K independent of the imbedding of K in ℂ. If ε is a unit in K of modulus 1, the it's a root of unity. Imbeddings are linear wrt. complex conjugation. If O is the ring of algebraic integers, n positive, and I an ideal of O solving |I|² = (n), then I is principal and has a generator with ε = n/|X|² a square of a real unit iff ∃ a solution Y ∈ O to |Y|² = n with (Y) = I. If I is principal, there is always a solution Z ∈ O of |Z|² = n² with (Z) = I².

Note the ideal group I'_P = {J ∈ I_P: J/J' ∈ G_{m, p}I_Q} containing I_P∩I⁺I_QH. If p is not self-conjugate modulo m = p^am', then I_P/I'_P ≅ (ℤ/wℤ) × (ℤ/uℤ)^e/2-1 where u = φ(p^a), e = φ(m')/ord_m'(p), w = u/w₀. If p is not self-conjugate modulo m = p^am', then C_PC⁺/C_QC⁺ ≅ (ℤ/2^δwℤ) × (ℤ/uℤ)^e/2-1, where δ ∈ {0, 1}; u, e and w as above, with w either 2 if p = 2, (p-1, m') if m' is even, or (p-1, 2m') if p and m' are odd. The relative class number h'_m of ℚ(ξ_m) is divisible by wu^e/2-1. Assume that f is a common divisor of f₁, ..., f_s with r_i^p-1 ≠ 1 (mod p²) if m = p^a > p. If the f_i are odd and for nonnegative integers d_i

A CM-field K with complex subfield k so that K/k is Galois, ℚ(ξ_m) with m ≠ 2 (mod 4) the largest cyclotomic field contained in K. Write D := Gal(K/k) with generators {σ}. Let the set of primes T of k with the ramification index of P ∈ T in K/k by R(P), then the ideal class group Cl_K of K contains a subgroup L isomorphic to Ω/Λ where Ω := ⊕_P∈T(ℤ/R(P)ℤ) and Λ is a subgroup of Ω of rank ≤ s and exponent dividing λ := lcm(f(m, σ_i), i = 1, ..., s). For a prime q, the subgroup of Cl_K generated by the primes in T_K has order of at least [Π_p∈T R(P)]/λ^s. Note d_q the q-rank of D, and R_q the number of R(P)s div by q, then the q-rank of Cl_K is ≥ R_q d_q. If m and |D| are odd and relative prime, then Cl_K contains a subgroup isomorphic to ⊕_P∈T(ℤ/R(P)ℤ). If K has a maximal real subfield K⁺ and ideal class group Cl_K, and r the number of finite primes ramified in K/K⁺, then d₂Cl_K ≥ r - 1.

An algebraic number field K and prime p, if d_pCl_K ≥ 2 + 2 √(d_pE_K + 1), K has an infinite p-class tower. If K has an ideal class group Cl_K and group of units E_K, and K/k a cyclic extension of degree p, then d_pCl_K ≥ ρ - d_pE_k - 1, where ρ is the number of finite and infinite primes of k ramified in K/k.