Abstract
We obtain new quantitative forms of Hilbert’s irreducibility theorem. In particular, we show that if f(X,T1,…,Ts) is an irreducible polynomial with integer coefficients, having Galois group G over the function field Q(T1,…,Ts), and K is any subgroup of G, then there are at most Of,ε(Hs−1+∣∣G/K∣∣−1+ε) specialisations t∈Zs with |t|≤H such that the resulting polynomial f(X) has Galois group K over the rationals.
Original language | English |
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Pages (from-to) | 1-14 |
Number of pages | 14 |
Journal | Acta Arithmetica |
Volume | 180 |
Early online date | 9 Aug 2017 |
DOIs | |
Publication status | E-pub ahead of print - 9 Aug 2017 |