## 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:

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