## The distribution of Galois groups and Hilbert’s irreducibility theorem.(English)Zbl 0484.12002

Assume $$f(\underline x,\underline t)$$ to be a polynomial over an algebraic number field $$K$$ in the indeterminates $$\underline x =(x_1,\ldots,x_r)$$, $$\underline t=(t_1,\ldots,t_s)$$. From Hilbert’s irreducibility theorem one knows that there are infinitely many specializations $$\underline t\to \underline \alpha \in K^s$$ such that $$f(\underline x,\underline \alpha)$$ is irreducible over $$K$$, provided $$f$$ was irreducible in $$K(x,t)$$. Moreover, in case $$r= 1$$, there are infinitely many specializations $$\underline t\to \underline \alpha \in K^s$$ such that the Galois group $$G(\alpha)$$ of $$f(x,\alpha)$$ coincides with the original group $$G$$ of $$f(x,\underline t)$$. In the paper under review, which in some sense is a continuation of the author’s paper in [Ill. J. Math. 23, 135–152 (1979; Zbl 0402.12005)], it is shown that $$G(\alpha) =G$$ is valid for almost all specializations; also, estimates are derived for the least modulus for which there exists a set of rational arithmetic progressions such that $$G(\alpha) = G$$ for $$\underline \alpha$$ in this progression. The nice thing is that the dependence of the estimates on the coefficients of $$f$$ turns out to be explicit and effective. Though the methods and proofs of the paper are very interesting and contain a lot of number theory, they seem to be too technical for being described within the frame of a review.

### MSC:

 11R45 Density theorems 11R09 Polynomials (irreducibility, etc.) 11R32 Galois theory 12E25 Hilbertian fields; Hilbert’s irreducibility theorem

Zbl 0402.12005
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