Abstract
Let K be a number field and \(K_\mathrm{ur}\) be the maximal extension of K that is unramified at all places. In a previous article (Kim, J Number Theory 166:235–249, 2016), the first author found three real quadratic fields K such that \(\mathrm {Gal}(K_\mathrm{ur}/K)\) is finite and non-abelian simple under the assumption of the generalized Riemann hypothesis (GRH). In this article, we extend the methods of Kim (2016) and identify more quadratic number fields K such that \(\mathrm {Gal}(K_\mathrm{ur}/K)\) is a finite nonsolvable group and also explicitly calculate their Galois groups under the assumption of the GRH. In particular, we find the first imaginary quadratic field with this property.
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Notes
To apply Schur’s lemma here, we have used that \(p\ne r,\) which is obvious, since \(p^2\) does not divide \(|\mathrm {GL}_2(\mathbb {F}_p)|.\)
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The second author was supported by the Israel Science Foundation (Grant No. 577/15).
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Kim, KS., König, J. Some examples of quadratic fields with finite non-solvable maximal unramified extensions II. Ramanujan J 51, 205–228 (2020). https://doi.org/10.1007/s11139-018-0046-3
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DOI: https://doi.org/10.1007/s11139-018-0046-3