Cyclotomic Fields in Finite Geometry 2025, 33 - The Fixing Theorem

For a solution X to the |X|² = n with n a positive integer, if σ∈Gal(ℚ(ξ_m)/ℚ) fixes all primes above n in ℚ(ξ_m), then X^σεX for some root of unity ε. Let K/k be a Galois extension of algebraic number fields, ℚ(ξ_m), m ≠ 2(mod 4) the largest cyclotomic field contained in K, and σ ∈ Gal(K/k), so that σ(ξ_m) = ξ^t_m and m = Π_p∈Sm_p where S is the set of primes dividing m. Define T_odd = {p∈S: p odd, t = 1(mod p), y_p > ord_mp(t)} and T is either T_odd∪{2} if t = 1 (mod 4) and y₂ > ord_m2(t), and just T_odd otherwise, and f(m,σ) is 2 gcd(m, Π_p∈Ty_p) if m is odd and y is even, m₂gcd(m, Π_p∈Ty_p) if m is even, t = 3 (mod 4) and 2m₂ divides t^y - 1, and gcd(m, Π_p∈Ty_p) otherwise. If X^σ = εX, X ∈ K, there is an m-th root of unity α and f(m, σ)-th root of unity η so that (Xα)^σ = η(Xα). (Xα)^{f(m, σ)} ∈ Fix σ.

Note G(p, e) := {G(χ): χ ∈ Γ} as the Gauss sum for p = ef + 1 an odd prime, with e ≠ 1, Γ the set of all characters of ℤ*_p of order dividing e. Let m = p^am', (p, m') = 1 and m ≠ 2 (mod 4). If X ∈ ℤ[ξ_m] is a solution of |X|² = p^b, b ≥ 1, then there is an integer j with Xξ^j_m ∈ ℤ[ξ_m'] equivalently X = ξ^j_mG(χ)Z, Z ∈ ℤ[ξ_m'], |Z|² = p^b-1, G(χ) ∈ G(p, w₀) where w₀ is either 2 for p = 2, (p - 1, m') if m' is even, and 2(p-1, m') if p and m' are odd. If X ∈ ℤ[ξ_m] is a solution for |X|² = n, where m = p^a, p is an odd prime and (n, p) = 1, n = Π^s_i=1 r_i^a_i and r_is distinct primes. If a ≥ 2, r_i^p-1 ≠ (mod p²) ∀ i. If f is a common divisor of ord_p(r_i), write p = ef + 1, then if n is square of a positive integer u, f > 2u(p-1)/p, then (X) = (u).

m, n positive integers, m = Π^t_i=1p_i^c_i the prime power decomposition of m. For each prime divisor q of n, m_q is Π_pi≠qp_i if m is odd or q = 2, and otherwise 4Π_{pi ≠ 2, q}p_i. If D(n) is the set of prime divisors of n, define F(m, n) = Π^t_i=1p_i^b_i the minimum multiple of Πⁱ_i=1p_i with all pairs (i, q), q ∈ D(n), then either q = p_i and (p_i, b_i) ≠ (2, 1), b_i = c_i, or q ≠ p_i and q^ord_mq(q) ≠ 1 (mod p_i^b_i+1). If for |X|² = n, X ∈ ℤ[ξ_m] where n, m positive integers, then Xξ^j_m ∈ ℤ[ξ_{F(m, n)}] for some j. Let X ∈ ℤ[ξ_m] of the form X = ∑_i=0^m-1 a_qξⁱ_m with 0 ≤ a_i ≤ C for C constant. if n is an integer, then