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 \(\Gamma_n\).


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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