Abstract
The two quadratic fields \({\Bbb Q}(\sqrt d )\) and \({\Bbb Q}(\sqrt {-3d} )\) are known to be closely related. A. Scholz [Sc] gave the “Spiegelung” relation between the 3-ranks of their ideal class groups. In Y. Kishi [Kil], we gave a precise Spiegelung relation by a constructive approach: every cubic polynomial which generates a cyclic cubic extension E of \({\Bbb Q}(\sqrt d )\) unramified outside 3 over \({\Bbb Q}(\sqrt d )\) with Gal(E/ℚ) ≅ S 3 is constructed by making use of an element of the associated field \({\Bbb Q}(\sqrt {-3d} )\) ; here S3 is the symmetric group on three symbols. In 1972, on the other hand, G. Gras [Grl] showed that all S 3-extensions of ℚ containing a fixed quadratic field \({\Bbb Q}(\sqrt d )\) correspond to some elements of \({\Bbb Q}(\sqrt {-3d} )\) (see Example 3.4 (1)). In this paper, we extend it for an arbitrary odd prime p over an algebraic number field k with k∩ℚ(ζ) = ℚ, where ζ is a primitive p-th root of unity (see Section 2).
This author was supported by JSPS Research Fellowships for Young Scientists.
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Imaoka, M., Kishi, Y. (2004). On Dihedral Extensions and Frobenius Extensions. In: Hashimoto, Ki., Miyake, K., Nakamura, H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0249-0_10
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DOI: https://doi.org/10.1007/978-1-4613-0249-0_10
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