Lattice Models of Finite Fields

作     者：Lucian M. Ionescu[lmiones@ilstu.edu] Mina M. Zarrin 

作者机构：Mathematics Department Illinois State University Normal IL USA 

出 版 物：《Advances in Pure Mathematics》 (理论数学进展（英文）)

年 卷 期：2017年第7卷第9期

页      面：451-466页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Finite Fields Algebraic Number Fields Ramification Theory Frobenius Element Congruence Zeta Function Weil Zero 

摘      要：Finite fields form an important chapter in abstract algebra, and mathematics in general, yet the traditional expositions, part of Abstract Algebra courses, focus on the axiomatic presentation, while Ramification Theory in Algebraic Number Theory, making a suited topic for their applications, is usually a separated course. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a much larger audience, and bridging the above mentioned gap. Such lattice models of finite fields provide a good basis for later developing their study in a more concrete way, including decomposition of primes in number fields, Frobenius elements, and Frobenius lifts, allowing to approach more advanced topics, such as Artin reciprocity law and Weil Conjectures, while keeping the exposition to the concrete level of familiar number systems. Examples are provided, intended for an undergraduate audience in the first place.