zbMATH — the first resource for mathematics

On the Hasse principle for cubic surfaces. (English) Zbl 0151.03405
E. S. Selmer has conjectured [Math. Scand. 1, 113–119 (1953; Zbl 0051.03202)] that every diagonal cubic form in four variables which represents 0 in every \(p\)-adic field represents 0 in the rational field \(\mathbb Q\). The authors disprove the conjecture by the example \(5x^3+12y^3+9z^3+10t^3\). The proof that this form does not represent 0 in \(\mathbb Q\) uses properties of the group of ideal classes of the field \(\mathbb Q\left(\root 3\of {30}, \root 3\of {90}\right)\). The necessary information about the class number of \(\mathbb Q\left(\root 3\of a, \root 3\of b\right)\) is conveniently gathered in an appendix.

11D25 Cubic and quartic Diophantine equations
14G05 Rational points
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
PDF BibTeX Cite
Full Text: DOI
[1] Jacobi, Canon Arithmeticus (1956)
[2] Hilbert, Jahresbericht der DMV 4 pp 175– (1897)
[3] Dirichlet, Vorlesungen über Zahlentheorie. Mit Zusdtzen versehen von B. Dedekind (1894)
[4] Brauer, Math. Nachr 4 pp 158– (1950) · Zbl 0042.03801
[5] Berwick, Integral Bases. Cambridge Tracts in Mathematics and Mathematical Physics (1927) · JFM 53.0142.01
[6] Swinnerton-Dyer, Mathematika 9 pp 54– (1962)
[7] Kuroda, Nagoya Math. J. 1 pp 1– (1950) · Zbl 0037.16101
[8] Selmer, Math. Scand 1 pp 113– (1953) · Zbl 0051.03202
[9] Selmer, Avh. Norske Vid. Akad. Oslo I pp 38– (1955)
[10] DOI: 10.1007/BF02940658 · Zbl 0007.10303
[11] DOI: 10.1112/jlms/s1-40.1.149 · Zbl 0124.02605
[12] Mordell, Pub. Math. Debrecen 1 pp 1– (1949)
[13] DOI: 10.1007/BF01187940 · Zbl 0066.39702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.