We show that irreducibility of a polynomial in any number of variables over a number field is a diophantine condition, i.e. captured by an existential formula. This generalises a previous result by Colliot-Thélène and Van Geel that the set of non-nth-powers is diophantine for any n. Our method is heavily based on the Brauer group, originating from Poonen's use of quaternion algebras as a technical tool for first-order definitions in number fields.

Siegel proved the finiteness of the set of solutions to the unit equation in a number ring, i.e., for a number field K with ring of integers O, the equation x+y=1 has only finitely many solutions in O*. That is, reformulated in more algebro-geometric terms, the hyperbolic curve P^1-{0,1,infinite} has only finitely many 'integral points'. In 1983, Faltings proved the Mordell conjecture generalizing Siegel's theorem: a hyperbolic complex algebraic curve has only finitely many integral points. Inspired by Faltings's and Siegel's finiteness results, Lang and Vojta formulated a general finiteness […]

Gauss a donné des formules pour le nombre de points entiers primitifs de la 2-sphère de rayon au carré égal à n. Ces formules sont en termes de nombres de classes d'anneaux quadratiques de discriminant étroitement liés à n. Cela mène à la question de savoir si ceci peut être expliqué par une action libre et transitive du groupe de Picard de cet anneau sur l'ensemble des tels points entiers primitifs à symétries globales SO_3(Z) près. Ceci est en effet le cas, et cette action peut être explicitée. L'outil utilisé […]