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Realising wreath products of cyclic groups as Galois groups. (English) Zbl 0662.12010

Let \(K\) be a field of characteristic zero, \(T\), \(X\) indeterminates algebraically independent over \(K\). For \(n\geq 1\), let \(k(n)\ge 2\) be an integer, \(f_n(X,T)=X^{k(n)}+T\). Put \(F_1(X,T)=f_1(X,T)\) and define for \(n\geq 1\), \(F_{n+1}(X,T)=F_n(f_{n+1}(X,T),T)\).
In Theorem 1 the author proves that if \(\overline{K}\) is the algebraic closure of \(K\) then the Galois group of \(F_n(X,T)\) over \(\overline{K}(T)\) is isomorphic to the wreath product \(\Gamma_n = G_1 [\ldots [G_n]\ldots]\) where for each \(i\le n\), \(G_i\) is the cyclic group of order \(k(i)\) with its natural permutation action on the symbols \(1,\ldots,k(i)\).
As a corollary the author proves that if \(K\) is a Hilbertian field containing the \(k(i)\)-th roots of \(1\) for \(i\le n\) then given \(t>1\) there is a finite Galois extension \(L\) over \(K\) such that \(\mathrm{Gal}(L/K)\) is isomorphic to the direct product of \(t\) copies of \(\Gamma_n\).

MSC:

11R32 Galois theory
12F12 Inverse Galois theory
20F29 Representations of groups as automorphism groups of algebraic systems
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
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References:

[1] Lang, Algebra (1965)
[2] Hall, The Theory of Groups pp 81– (1959)
[3] Grothendieck, Lecture Notes in Maths pp 224– (1971)
[4] Fried, Field Arithmetic, Ergebnisse der Math., 3{\(\deg\)} Folge (1986)
[5] Lang, Fundamentals of Diophantine Geometry (1983)
[6] DOI: 10.2996/kmj/1138036122 · Zbl 0435.12010
[7] Šafarevifč, Izv. Akad Nank SSSR, Ser. Mat. 18 pp 26– (1954)
[8] DOI: 10.1112/plms/s3-51.3.385 · Zbl 0622.12011
[9] Zariski, Commutative Algebra (1958)
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