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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.

##### MSC:
 11D25 Cubic and quartic Diophantine equations 14G05 Rational points 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
##### Keywords:
cubic form; Hasse principle; cubic surfaces
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##### References:
 [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
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