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A359291
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Absolute discriminants of imaginary quadratic fields with elementary bicyclic 5-class group and capitulation type the identity permutation.
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2
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OFFSET
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1,1
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COMMENTS
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An algebraic number field with this capitulation type has a 5-class field tower of precise length 2 with Galois group isomorphic to the Schur sigma-group SmallGroup(3125,14). It is a solution to the problem posed by Olga Taussky-Todd in 1970.
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REFERENCES
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A. Azizi et al., 5-Class towers of cyclic quartic fields arising from quintic reflection, Ann. math. Québec 44 (2020), 299-328. (p. 314)
D. C. Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401-456. (Sec. 3.5.2, p. 448)
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LINKS
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EXAMPLE
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The first imaginary quadratic field with 5-class group (5,5) and identity capitulation (123456) has discriminant -89751 and was discovered by Daniel C. Mayer on 03 November 2011. It has ordinal number 31 in the sequence A359871 of all imaginary quadratic fields with 5-class group (5,5). The discriminant -89751 appears in the table on page 130 in the Ph.D. thesis of Tobias Bembom, 2012. However, contrary to his assertion in Remark 2 on page 129, his method was not able to detect the identity capitulation. Consequently, Bembom only found a (non-identity) permutation (135246) but did not solve Taussky's problem.
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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STATUS
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approved
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