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On number fields with power-free discriminant

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Abstract

Given a finite transitive permutation group G, we investigate number fields F/ℚ of Galois group G whose discriminant is only divisible by small prime powers. This generalizes previous investigations of number fields with squarefree discriminant. In particular, we obtain a comprehensive result on number fields with cubefree discriminant. Our main tools are arithmetic-geometric, involving in particular an effective criterion on ramification in specializations of Galois covers.

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Acknowledgements

I thank Peter Müller for some helpful comments on Proposition 2.3 and beyond. I thank the Department of Mathematics at KIAS (Seoul) for their hospitality during the time this article was written.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, Republic of Korea

    Joachim König

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  1. Joachim König
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Correspondence to Joachim König.

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König, J. On number fields with power-free discriminant. Isr. J. Math. 235, 413–437 (2020). https://doi.org/10.1007/s11856-020-1962-7

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  • DOI: https://doi.org/10.1007/s11856-020-1962-7

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