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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)\(\geq 2\) 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 \(\Gamma_ n=G_ 1[...[G_ n]...]\) where for each \(i\leq n\), \(G_ i\) is the cyclic group of order k(i) with its natural permutation action on the symbols 1,...,k(i).
As a corollary the author proves that if K is a Hilbertian field containing the k(i)-th roots of 1 for \(i\leq n\) then given \(t>1\) there is a finite Galois extension L over K such that Gal(L/K) is isomorphic to the direct product of t copies of \(\Gamma_ n\).
Reviewer: T.Soundararajan

11R32 Galois theory
12F10 Separable extensions, Galois theory
20F29 Representations of groups as automorphism groups of algebraic systems
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
Full Text: DOI
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