Splitting fields of $X^n-X-1$ (particularly for $n=5$), prime decomposition and modular forms

Chandrashekhar B. Khare, Alfio Fabio La Rosa, Gabor Wiese

We study the splitting fields of the family of polynomials $f_n(X)=X^n-X-1$. This family of polynomials has been much studied in the literature and has some remarkable properties.…