Geodesic
Algebraic points on quartic curves over function fields
Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 187-190

Voir la notice de l'article provenant de la source Cambridge University Press

The following general problem is of interest. Let Λ be an irreducible algebraic variety of degree d, in projective n-space Pn, defined over a field k; and suppose that K is a finite extension of k with [K: k] prime to d. If Λ has a point defined over K, then does it necessarily have a point defined over k?It has been studied in various instances by several authors: see, for example, Cassels [2], Coray [3, 4], Pfister [5], Bremner, Lewis, Morton [1]. Coray [3] shows that a quartic curve Λ over Q may possess points in extension fields of Q of every odd degree greater than one, but have no points in Q itself. Some further examples of this instance occur in the paper of Bremner, Lewis, Morton, with the additional property that the curve Λ also possesses points in every p-adic completion Qp of Q. 
Bremner, Andrew. Algebraic points on quartic curves over function fields. Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 187-190. doi: 10.1017/S001708950000598X
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[1] 1.Bremner, A., Lewis, D. J. and Morton, P., Some varieties with points only in a field extension, Archiv der Math. 43 (1984), 344–350. Google Scholar | DOI

[2] 2.Cassels, J. W. S., On a problem of Pfister about systems of quadratic forms, Archiv der Math. 33 (1979), 29–32. Google Scholar | DOI

[3] 3.Coray, D. F., Algebraic points on cubic hypersurfaces, Ada Arith. 30 (1976), 267–296. Google Scholar | DOI

[4] 4.Coray, D. F., On a problem of Pfister about intersections of three quadrics, Archiv der Math. 34 (1980), 403–411. Google Scholar | DOI

[5] 5.Pfister, A., Systems of quadratic forms, Colloque sur les formes quadratiques, 2 Bull. Soc.Math. France, Mem. No. 59, (1979), 115–123. Google Scholar

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