Cyclotomic Fields in Finite Geometry 2025, 32 - Some Number Theory

An algebraic integer for which all conjugates have absolute value 1 is a root of unity. For m a positive integer and p prime, m = p^am' with (m', p) = 1, a ≥ 0. p factors in ℚ(ξ_m). Write (p) = Π_i=1^tπ_i^φ(pa) with t = φ(m')/ord_m'(p) and π_i are distinct prime ideals. If p is also rational, and P a prime ideal above p in ℤ[ξ_m], and write m = p^am', (m', p) = 1. The decomposition group of P consists of all σ ∈ Gal(ℚ(ξ_m)/ℚ) with integer j such that σ(ξ_m') = ξ_m'^pj. For A ∈ ℤ[ξ_m] with |A|² = o mod t^2b, with b, t integers > 0 and t is self-conjugate modulo m. A = 0 mod t^b. Let r = p^a, χ a character of 𝔽*_r, then G(χ) = ∑_x∈𝔽*rχ(x)ξ_p^Tr(x). If χ nontrivial, then |G(χ)|² = r.

For p prime, q = p^a, π a prime ideal of ℚ(ξ_q-1) above p, π' a prime ideal of ℚ(ξ_q-1, ξ_p) above π. Note ν_π' as the π'-adic evaluation, ω = ω(π) the Teichmueller character of 𝔽*_q corresponding to π, then ν_π'(G(ω^j)) = S_p(j) for 1 ≤ j < q - 1.

For a prime power r and positive integer s, F = 𝔽_r, E = 𝔽_r^s, χ a character of F* with χ' of E* by χ'(x) = χ(N_E/F(x)) with N_E/F is the norm of E relative to F. It follows that G(χ') = (-1)^s-1G(χ)^s. If p is prime and b a positive integer, if (p, b) ≠ (2, 1), and s an integer satisfying s = 1 (mod p^b), s ≠ 1 (mod p^b+1), then ord_p^c(s) = p^c-b ∀ c ≥ b. If s and t are integers such that ord_p^b(s) = ord_p^b(t) is a power of p. Assume s = t = 1 (mod 4) if p = 2, then s and t generate the same subgroup of the multiplicative group ℤ*_p^b.

For p prime and G a finite abelian group with a cyclic Sylow p-subgroup S. If with Y ∈ ℤ[G] χ(Y) = 0 mod p^a ∀ characters χ of G of order divisible by [S], then there are X₁, X₂ ∈ ℤ[G] with Y = p^aX₁ + PX₂ where P is the unique subgroup of order p of G. If Y has only positive definite, then X₁, X₂ can be chosen to also have only positive definite coefficients.