We develop a Galois descent approach to finite-field Fourier spectra over an arbitrary finite base field. Let $\mathbb K=\mathbb F_q$ and $\mathbb L=\mathbb F_{q^m}$. If a Fourier transform is applied to a $\mathbb K$-valued vector, then its spectrum is not an arbitrary element of $\mathbb L^n$: it satisfies the Frobenius consistency relation \[ V_s^q=V_{qs \bmod n}. \] We prove a general Galois-descent theorem for Fourier transforms on finite abelian groups, characterize the one-dimensional spectra as products of subfields indexed by $q$-cyclotomic classes, and show that the orbit-seed representation is optimal in base-field coordinates. For arbitrary vectors in $\mathbb L^n$, we study a two-stage representation $g=f+h$, where $f$ is class-consistent and $h$ is a residual. The residual optimization separates over cyclotomic classes. We give exact support minimization, a symbol weight enumerator for the class-consistent code, recovery guarantees, global covering-radius formulas, random residual tail bounds, and entropy-type lower bounds. We also discuss implementation consequences for trace decompositions, normal bases, canonical subfield embeddings, and sparse-polynomial residual backends.

翻译：针对任意有限基域上的有限域傅立叶谱，我们发展了一种伽罗瓦下降方法。设$\mathbb K=\mathbb F_q$，$\mathbb L=\mathbb F_{q^m}$。若对$\mathbb K$值向量应用傅立叶变换，其谱并非$\mathbb L^n$中的任意元素，而是满足Frobenius相容关系：\[ V_s^q=V_{qs \bmod n}。\] 我们证明了有限阿贝尔群上傅立叶变换的一般伽罗瓦下降定理，将一维谱刻画为由$q$-循环分圆类索引的子域乘积，并证明轨道-种子表示在基域坐标下是最优的。对于$\mathbb L^n$中的任意向量，我们研究了两阶段表示$g=f+h$，其中$f$是类相容的，$h$为残差。残差优化在循环分圆类上可分离。我们给出了精确支撑极小化、类相容码的符号权值枚举器、恢复保证、全局覆盖半径公式、随机残差尾部界以及熵型下界。此外，我们还讨论了迹分解、正规基、典范子域嵌入以及稀疏多项式残差后端等实现层面的影响。