{'fraction_non_alphanumeric': 0.015895953757225433, 'fraction_numerical': 0.000722543352601156, 'mean_word_length': 5.382488479262673, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Galois theory of Artinian simple module algebras

This main purpose of this article is the unification of the Galois theory of algebraic differential equations by Umemura and the Galois theory of algebraic difference equations by Morikawa-Umemura in a common framework using Artinian simple D-module algebras, where D is a bialgebra. We construct the Galois hull of an extension of Artinian simple D-module algebras and define its Galois group, which consists of infinitesimal coordinate transformations fulfilling certain partial differential equations and which we call Umemura functor. We eliminate the restriction to characteristic 0 from the above mentioned theories and remove the limitation to field extensions in the theory of Morikawa-Umemura, allowing also direct products of fields, which is essential in the theory of difference equations. In order to compare our theory with the Picard-Vessiot theory of Artinian simple D-module algebras due to Amano and Masuoka, we first slightly generalize the definition and some results about them in order to encompass as well non-inversive difference rings. Finally, we give equivalent characterizations for smooth Picard-Vessiot extensions, describe their Galois hull and show that their Umemura functor becomes isomorphic to the formal scheme associated to the classical Galois group scheme after a finite \'etale base extension.