Abstract
The order-reversing bijection between field extensions and subgroups of the Galois group G follows from the equivalence between the opposite of the category of étale algebras and the category of discrete G-spaces [2]. We show that the basic ingredient for this equivalence of categories, and for various known generalizations, is a factorization system for variable categories.
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Janelidze, G., Schumacher, D. & Street, R. Galois theory in variable categories. Appl Categor Struct 1, 103–110 (1993). https://doi.org/10.1007/BF00872989
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DOI: https://doi.org/10.1007/BF00872989