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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.

  • Skew braces and the Yang-Baxter equation, and their relationship with Hopf-Galois structures.

  • 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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Charla USC.jpg

Photo: Me giving a talk at Santiago de Compostela, March 2024

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Research papers:

  1. D. Gil-Muñoz. "A generalization of Kummer theory to Hopf-Galois extensions". J. Algebra 660 (2024), pages 190-235. DOI: 10.1016/j.jalgebra.2024.07.017. arXiv:2305.08648

  2. D. Gil-Muñoz, M. Tinková. "The lifting problem for universal quadratic forms over simplest cubic fields". Bull. Aust. Math. Soc. 110 (2024), pages 77-89. DOI: 10.1017/S0004972723000953

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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:

  1. D. Gil-Muñoz. "Hopf-Galois module structure of degree p extensions of p-adic fields". arXiv:2410.10383.

  2. T. Crespo, D. Gil-Muñoz, A. Rio, M. Vela. "Determining skew left braces of size np". arXiv:2302.13098.

  3. D. Gil-Muñoz, M. Tinková. "Additive structure of non-monogenic simplest cubic fields". arXiv:2212.00364.

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Other publications:

  1. 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

  2. "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:

  1. 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).

  2. Final master project: Explicit class field theory via elliptic curves. Advisor: Xavier Guitart (University of Barcelona).

  3. Final degree project: Anillos numéricos (in Spanish). Advisor: María de los Ángeles Gómez Molleda (University of Malaga).

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