Principal factors in a pure cubic field and its unramified cyclic cubic extensions. (English) Zbl 0618.12005
Let m be a positive 3rd power free integer and let \(L={\mathbb{Q}}(^ 3\sqrt{m})\). Denote by \(p_ 1,...,p_ t\) the rational primes which are fully ramified in L. A positive integer \(d=p_ 1^{e_ 1}...p_ t^{e_ t}\) is called a principal factor for L if it is not of the form \(m^{f_ 0}\cdot p_ 1^{f_ 1}...p_ t^{f_ t}\) and if the principal ideal of L generated by d is the 3rd power of a principal ideal of L. The author describes conditions in terms of units and genus theory which are equivalent to the existence of a principal factor \(\equiv 1 mod 9\). These are partly analoguous to the real quadratic case.
Reviewer: H.Opolka
References:
| [1] | Barrucand, P.; Cohn, H., Remarks on principal factors in a relative cubic field, J. Number Theory, 3, 226-239 (1971) · Zbl 0218.12002 · doi:10.1016/0022-314X(71)90040-0 |
| [2] | Fröhlich, A., The genus field and genus group in finite number fields II, Mathematika, 6, 142-146 (1969) · Zbl 0202.33005 |
| [3] | Gerth, F., On 3-class groups of pure cubic fields, J. reine angew. Math., 278/279, 52-62 (1975) · Zbl 0334.12011 |
| [4] | Hasse, H., Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper (1970), Würzburg/Wien: Physica-Verlag, Würzburg/Wien · JFM 52.0150.19 |
| [5] | Iimura, K., On the unit groups of certain sextic number fields, Abh. Math. Sem. Univ. Hamburg, 50, 32-39 (1980) · Zbl 0416.12002 · doi:10.1007/BF02941412 |
| [6] | Nakagoshi, N., The structure of the multiplicative group of residue classes modulop^N+1, Nagoya Math. J., 73, 41-60 (1979) · Zbl 0393.12023 |
| [7] | Odai, Y., Some unramified cyclic cubic extensions of pure cubic fields, Tokyo J. Math., 7, 391-398 (1984) · Zbl 0576.12005 · doi:10.3836/tjm/1270151734 |
| [8] | Scholz, A., Idealklassen und Einheiten in kubischen Körpern, Monatshefte Math. Phys., 40, 211-222 (1933) · Zbl 0007.00301 · doi:10.1007/BF01708865 |
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