A note on the Hilbert irreducibility theorem. The irreducibility theorem and the strong approximation theorem
Yasuo Morita · Proceedings of the Japan Academy Series A Mathematical Sciences · 1990
Introduction.Let k be a finite algebraic number field.For any irre- ducible polynomial f(t, x)e k(t)[x], let U, denote the set consisting of all s e k such that f(s, x) is defined and irreducible in k[x].A subset of k o this form is called a basic Hilbert subset of k.Further, an intersection o a non-empty Zariski open subset of k and a finite number of basic Hilbert subsets of k is called a Hilbert subset of k.In this paper, we obtain the ollowing theorem" Main theorem.Let f2 be the set of all primes of a finite algebraic number field k, let q be an element of , and let S be a finite subset of such that 2-S-{q} contains only non-archimedean primes of k.We choose an element of k for each p e S.Then, for any positive number and for any Hilbert subset H of k, there exists an element e H such that {,,o--o] foranypeS, [_1 for any p e -S-{q}.Clearly, this theorem shows that the Hilbert irreducibility theorem and the strong approximation theorem for k are compatible.It is easy to re- duce this theorem to the Hilbert irreducibility theorem if S contains only non-archimedean primes, but it seems nontrivial if S contains archimedean primes.We prove the theorem by modifying an argument in S. Lang [1], VIII, 1.The author would like to thank Professor Peter Roquette for valuable comments.1. Hilbert sets and rational points of algebraic curves.Let k be a finite algebraic number field, and let H be a Hilbert subset of k.Then, or svme non-empty Zariski open subset O ot k, we can write ((= U,,), where f(t, x) is an irreducible polynomial in k(t)[x] and U,, is the basic Hilbert subset corresponding to f.Here, by multiplying an ele- ment o k[t] and changing O if necessary, we may assume ft(t, x) e k[t, x].Let f(t, x) be one of the f(t, x).Let k(t) be the algebraic closure o k(t), and write f(t, x)=a(t) [=1 (x--a) (a(t) e k[t], a e k(t)).Let f(t, x)= *)Dedicated to Professor Ichiro S.t.TAKE on his sixtieth birthday.