Abstract
Let p be a prime number, and let \({K}\) be a number field. For \(p=2\), assume moreover that \({K}\) is totally imaginary. In this note, we prove the existence of asymptotically good extensions \({L}/{K}\) of cohomological dimension 2 in which infinitely many primes split completely. Our result is inspired by a recent work of Hajir, Maire, and Ramakrishna.
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The authors thank Karim Belabas, Baptiste Cerclé, Farshid Hajir, and Alexander Schmidt for useful comments; and Philippe Lebacque and Jan Mináč for their interests in this work. They also want to thank the anonymous referee for his/her careful reading of the paper. CM was partially supported by the ANR project FLAIR (ANR-17-CE40-0012), and by the EIPHI Graduate School (ANR-17-EURE-0002).
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Hamza, O., Maire, C. A note on asymptotically good extensions in which infinitely many primes split completely. Arch. Math. 115, 523–534 (2020). https://doi.org/10.1007/s00013-020-01500-0
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DOI: https://doi.org/10.1007/s00013-020-01500-0