{"paper":{"title":"Cyclotomic finite-field Fourier spectra: Galois descent, native subfields, and residual coding","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.GR","math.IT"],"primary_cat":"math.AC","authors_text":"Anton Zarubin, Daniil Sizikov, David Kumallagov","submitted_at":"2026-05-19T16:20:13Z","abstract_excerpt":"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 repres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20062","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.20062/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}