STNB2026 (39a edició)
Hopf-Galois theory and applications to number theory
Coordinació
Daniel Gil Muñoz i Cornelius Greither
Descripció
In this course we offer a detailed introduction to Hopf-Galois theory, which is a generalization of Galois theory involving the use of Hopf algebras. We shall start with the study of Hopf algebras and their duals, as well as their actions on rings, introducing the equivalent notions of Hopf-Galois extensions, which are generalizations of Galois extensions, and Hopf-Galois objects, which can be regarded as their dual counterpart. The former notion involves Hopf-Galois structures on a field extension, defined as pairs of a Hopf algebra and a linear action on the extension. Among the very basics of Hopf-Galois theory, there is the Greither-Pareigis theorem, which establishes a bijective correspondence between the Hopf-Galois structures on a separable extension and a subclass of permutation subgroups of $n$ letter, where $n$ is the degree of the extension. The study of the Hopf-Galois structures using these techniques is the so-called Greither-Pareigis theory, of which we shall also see some applications. Finally, we shall study the use of Hopf-Galois extensions in number theory, and more concretely in studying the module structure of the ring of integers in extensions of number fields and $p$-adic fields.
Xerrades
Referències
S. U. Chase and M. E. Sweedler. Hopf Algebras and Galois Theory. 1st ed. Lecture Notes in Mathematics. Springer, 1969. <\p>
L. N. Childs. Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory. 1st ed. Mathematical Surveys and Monographs 80. American Mathematical Society, 2000. isbn: 0-8218-2131-8. <\p>
L. N. Childs, C. Greither, K. P. Keating, A. Koch, T. Kohl, P. J. Truman, and R. G. Underwood. Hopf Algebras and Galois Module Theory. 1st ed. Mathematical Surveys and Monographs 260. American Mathematical Society, 2021. <\p>
C. Greither and B. Pareigis. “Hopf Galois theory for separable field extensions”. In: Journal of Algebra 106.1 [1987], pp. 239–258. issn: 0021-8693. doi: https://doi.org/10.1016/0021- 8693(87)90029-9. <\p>
S. Montgomery. Hopf algebras and their actions on rings. Conference Board of the Mathematical Science. American Mathematical Society, 1993. <\p>
P. J. Truman. Hopf-Galois Module Structure of Some Tamely Ramified Extensions. PhD thesis. University of Exeter, 2009. <\p>
R. G. Underwood. Fundamentals of Hopf Algebras. 1st ed. Universitext. Springer, 2015. isbn: 978-3-319-18990-1. doi: 10.1007/978-3-319-18991-8. <\p>
