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On the Galois group of the generalized Fibonacci polynomial. (English) Zbl 1034.12003

For an integer \(n\geq 2\), define the \(n\)th generalized Fibonacci polynomial by \(f_n(x):= x^n- x^{n-1}-\dots- x-1\). D. Wolfram [Fibonacci Q. 36, 129–145 (1998; Zbl 0911.11014)] conjectured that the Galois group \(G_n\) of \(f_n\) over the field of rational numbers is the symmetric group \(S_n\). Here the authors prove that \(G_n\) is not contained in the alternating group \(A_n\) and that it is not 2-nilpotent.

MSC:

12F10 Separable extensions, Galois theory
12E10 Special polynomials in general fields

Citations:

Zbl 0911.11014
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