My research interests lie mostly in Algebraic Number Theory and its interactions with (Hopf-) Galois theory. My current lines of research can be summarized in​:
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Module structure of the ring of integers of an extension of local or global fields.
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Skew braces and the Yang-Baxter equation, and their relationship with Hopf-Galois structures.
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Bounds for the rank (number of variables) of universal quadratic forms over a totally real number field and indecomposable integers.
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Besides, I am interested in learning about arithmetic geometry, and more concretely topics like elliptic curves or the theory of schemes.
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Photo: Me giving a talk at Vilanova i la Geltrú, 2019
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Research papers:
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D. Gil-Muñoz, M. Tinková. "The lifting problem for universal quadratic forms over simplest cubic fields". Bull. Aust. Math. Soc. DOI: 10.1017/S0004972723000953
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T. Crespo, D. Gil-Muñoz, A. Rio, M. Vela. "Inducing braces and Hopf Galois structures". J. Pure Appl. Algebra 227 (2023), 107371. DOI: 10.1016/j.jpaa.2023.107371. arXiv:2206.03810.
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D. Gil-Muñoz. "The ring of integers of Hopf-Galois degree p extensions of p-adic fields with dihedral normal closure". J. Number Theory 245 (2023), pages 65-118. DOI: 10.1016/j.jnt.2022.10.002. arXiv:2205.13517.
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T. Crespo, D. Gil-Muñoz, A. Rio, M. Vela. "Left braces of size 8p". J. Algebra 617 (2023), pages 317-339. DOI: 10.1016/j.jalgebra.2022.11.011. arXiv:2205.04201.
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D. Gil-Muñoz, A. Rio. "Hopf-Galois module structure of quartic Galois extensions of Q". J. Pure Appl. Algebra 266 (2022), 107045. DOI: 10.1016/j.jpaa.2022.107045. arXiv:2107.13515.
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D. Gil-Muñoz, A. Rio. "Induced Hopf Galois structures and their Local Hopf Galois Modules". Pub. Mat. 66 (2022), pages 99-128. DOI: 10.5565/PUBLMAT6612204. arXiv:1910.06083.
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Preprints:
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D. Gil-Muñoz. "A generalization of Kummer theory to Hopf-Galois extensions". arXiv:2305.08648
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T. Crespo, D. Gil-Muñoz, A. Rio, M. Vela. "Double semidirect products and skew left braces of size np". arXiv:2302.13098.
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D. Gil-Muñoz, M. Tinková. "Additive structure of non-monogenic simplest cubic fields". arXiv:2212.00364.
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Other publications:
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D. Gil-Muñoz. "Constructing normal integral bases of Hopf Galois extensions". TEMat monográficos, 2 (2021): Proceedings of the 3rd BYMAT Conference, pages 47-50. issn: 2660-6003. url: https://temat.es/monograficos/article/view/vol2-p47
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"D. Gil-Muñoz. El duodécimo problema de Hilbert para cuerpos cuadráticos imaginarios". TEMat, 2 (2018), pages 15-30. issn: 2530-9633. url: https://temat.es/articulo/2018-p15
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Downloads:
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PhD thesis: An effective method to study the Hopf-Galois module structure of certain extensions of fields. Advisors: Anna Rio (Politechnical University of Catalonia) and Teresa Crespo (University of Barcelona).
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Final master project: Explicit class field theory via elliptic curves. Advisor: Xavier Guitart (University of Barcelona).
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Final degree project: Anillos numéricos (in Spanish). Advisor: María de los Ángeles Gómez Molleda (University of Malaga).
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