FLINT3.7.0-dev
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fq_embed_matrices

Synopsis#include <flint/fq_embed.h>
void fq_embed_matrices(fmpz_mod_mat_t embed, fmpz_mod_mat_t project, const fq_t gen_sub, const fq_ctx_t sub_ctx, const fq_t gen_sup, const fq_ctx_t sup_ctx, const fmpz_mod_poly_t gen_minpoly)source

Given the output of fq_embed_gens for fields kk = sub_ctx and KK = sup_ctx of degrees m∣nm \mid n, computes the n×mn \times m matrix embed of the embedding φ:k→K\varphi: k \to K that maps gen_sub to gen_sup, and an m×nm \times n matrix project of a kk-linear projection π:K→k\pi: K \to k with π∘φ=id\pi \circ \varphi = \mathrm{id}, so that project⋅embed\mathit{project} \cdot \mathit{embed} is the m×mm \times m identity. The matrices act on column vectors of coordinates in the monomial bases.

The projection is π(y)=φ−1(Tr⁡K/φ(k)(θy))\pi(y) = \varphi^{-1}(\operatorname{Tr}_{K/\varphi(k)}(\theta y)) for an element θ\theta with Tr⁡K/φ(k)(θ)=1\operatorname{Tr}_{K/\varphi(k)}(\theta) = 1: θ=1/(n/m)\theta = 1/(n/m) if p∤n/mp \nmid n/m, and otherwise θ=xi/Tr⁡(xi)\theta = x^i / \operatorname{Tr}(x^i) for a suitable power of the generator. The trace is computed with fq_embed_trace_matrix.

Cite this page

The FLINT team, “fq_embed_matrices (fq_embed.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/fq_embed/fq_embed_matrices.html (generated 2026-11-02).