This review presents a comprehensive comparative analysis of classical and modern approaches to algebraic functions over finite fields. We trace the historical evolution from Dedekind and Weber through Hasse, Weil, and Chevalley to Christol and Kedlaya, examining how each period contributed foundational insights. The classical framework, rooted in algebraic function field theory and algebraic geometry, is systematically contrasted with modern approaches based on automatic sequences and Hahn series. Key results including Christol’s theorem (1980), Kedlaya’s generalization (2006), the Hurwitz genus formula, and the Riemann\u2013Roch theorem are discussed with illustrative examples. Applications in coding theory, cryptography, and emerging research directions\u2014including post-quantum cryptography, 5G\/6G communications, DNA storage, and quantum error correction\u2014are surveyed. The paper concludes with an analysis of open problems and possible future research directions, providing a structured overview for researchers interested in the intersection of algebraic geometry, automata theory, and information theory. This review aims to integrate the historical evolution, the classical geometric framework, and the modern automata-theoretic perspective into a unified comparative presentation while highlighting current research trends and open problems. Illustrative examples are provided in a dedicated appendix to demonstrate key concepts through explicit algebraic computations and automata constructions. Specifically, this review analyzes 65 selected publications spanning the historical period from 1882 to 2024 to contextualize these theoretical shifts.<\/p>\n\n\n\n
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