An effective version of Hilbert’s irreducibility theorem. (English) Zbl 0729.12005
Sémin. Théor. Nombres, Paris/Fr. 1988-89, Prog. Math. 91, 241-249 (1990).
[For the entire collection see Zbl 0711.00009.]
The author gives an effective proof of Hilbert’s irreducibility theorem. The technique is used also to make some explicit calculations, obtaining an explicit example of a cubic polynomial with integral coefficients defining a smooth cubic surface X over \({\mathbb{Q}}\) such that the action of the Galois group of \({\mathbb{Q}}\) on \(Pic(X_{{\mathbb{Q}}})\) is the largest possible.
The author gives an effective proof of Hilbert’s irreducibility theorem. The technique is used also to make some explicit calculations, obtaining an explicit example of a cubic polynomial with integral coefficients defining a smooth cubic surface X over \({\mathbb{Q}}\) such that the action of the Galois group of \({\mathbb{Q}}\) on \(Pic(X_{{\mathbb{Q}}})\) is the largest possible.
Reviewer: C.-P.Ionescu (Bucureşti)
MSC:
| 12E25 | Hilbertian fields; Hilbert’s irreducibility theorem |
| 14C22 | Picard groups |
| 14A05 | Relevant commutative algebra |
| 14Q20 | Effectivity, complexity and computational aspects of algebraic geometry |
