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GALOIS THEORY OF COMMUTATIVE RINGS: A HISTORICAL PERSPECTIVE

This paper is intended as an introduction to the Galois theory of commutative rings. Building on the framework provided by Rognes [9], we explain a generalization of the notion of Galois extension from fields to commutative rings and summarize some important properties of such extensions. Analogous to the Fundamental Theorem of Galois Theory for fields, given a G-Galois extension T/R of commutative rings, there is a bijection between separable R-subalgebras of T and subgroups H ≤ G. Applying this theory to number field extensions L/K lets us understand the structure of the number ring extension OL /OK in terms of how L ramifies over K.