## 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

### Keywords:

Galois group; Fibonacci polynomial

Zbl 0911.11014
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