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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

Citations:

Zbl 0402.12005
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