Research proof index

At most two packed indices share a residue class mod d/2k. Given a residue r with j ≡ r (mod d/2k = 2^{α-κ-1}), the index j : Fin (d/2^κ) is either the first-half witness (value r) or the second-half witness (value d/2k + r). Combined with packExp_mod_eq, this pins down exactly which (at most two) packed summands can contribute to a give...

Factor existence. For q ≡ 5 (mod 8), X^{2^α}+1 factors over Z_q as a product of two coprime irreducible polynomials. Extracted from cyclotomic_card_normalizedFactors (the two normalized factors), with coprimality from Irreducible.coprime_iff_not_dvd.

fixedBasisMap is injective: distinct coefficient vectors give distinct fixed elements, read off at coefficient position (d/2k)·s via vElt_coeff and cancellation of the unit 2 if s=0 else 1.

Support of a fixed element (Eq. 7). Every x ∈ R_q^H has its nonzero coefficients on multiples of d/2k = 2^{α-κ-1}: for p < d with 2^{α-κ-1} ∤ p, x.1.coeff p = 0. This comes from the symmetric vElt basis (fixedBasisMap is surjective by card_fixedSubring_eq), each of whose elements is supported , by vElt_coeff_full , on the two multipl...

R_q^H is a field (Hachi [NOZ26, §3, Lemma 5], field part). Since R_q^H ⊆ R_q^{σ_{-1}} (fixedSubring_le_conjFixedSubring) and the latter is a field (conjFixedSubring_isField), R_q^H is a finite integral domain, hence a field. Proven modulo conjFixedSubring_isField: the inclusion R_q^H ↪ R_q^{σ_{-1}} is an injective ring hom into a field,...

Hachi [NOZ26, §3, Lemma 5]: R_q^H ≅ F_{q^k}. For q ≡ 5 (mod 8) and 2·2^κ ∣ 2^α, the fixed subring is ring-isomorphic to GaloisField q (2^κ) (= F_{q^{2^κ}}). Combines fixedSubring_isField with card_fixedSubring_eq (|R_q^H| = q^{2^κ}) and the classification of finite fields by cardinality. Proven modulo conjFixedSubring_isField (via `f...