A359291 Absolute discriminants of imaginary quadratic fields with elementary bicyclic 5-class group and capitulation type the identity permutation.

89751, 235796, 1006931, 1996091, 2187064

Offset: 1

Author

Daniel Constantin Mayer, Dec 24 2022

Keywords

• nonn
• more
• hard

Comments

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.

Examples

			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.
		

References

• 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)

Links

• A. Azizi et al., 5-Class towers of cyclic quartic fields arising from quintic reflection, arXiv:1909.03407 [math.NT], 2019.
• T. Bembom, The capitulation problem in class field theory, Dissertation, Univ. Göttingen, 2012. (Sec. 6.3, p. 128)
• D. C. Mayer, The distribution of second p-class groups on coclass graphs, arXiv:1403.3833 [math.NT], 2014.
• O. Taussky-Todd, A remark concerning Hilbert's Theorem 94, J. reine angew. Math. 239/240 (1970), 435-438.

Crossrefs

Cf. A359296.