论文标题
$ x^n-x-1 $的分组字段（尤其是$ n = 5 $），主要分解和模块化表格
Splitting fields of $X^n-X-1$ (particularly for $n=5$), prime decomposition and modular forms
论文作者
论文摘要
我们研究多项式$ f_n（x）= x^n-x-1 $的分裂场。这个多项式家族在文献中已经进行了大量研究，并且具有出色的特性。 Serre与Primes $ n_p（f_n）$的功能相关，对于固定的$ n \ leq 4 $和$ p $ A a Varying Prime的功能，该功能计算了$ f_n（x）$的根数（x）$ in $ \ mathbb f_p $与模块化形式的系数。我们研究了$ n = 5 $的情况，并将$ n_p（f_5）$与$ 5 $ 5 $模块化表单相关联，而不是$ \ mathbb q $，并且与特征0，平行重量1 Hilbert Modular Forms超过$ \ MATHBB Q（\ sqrt {19 \ cdot 151}）$。
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. Serre related the function on primes $N_p(f_n)$, for a fixed $n \leq 4$ and $p$ a varying prime, which counts the number of roots of $f_n(X)$ in $\mathbb F_p$ to coefficients of modular forms. We study the case $n=5$, and relate $N_p(f_5)$ to mod $5$ modular forms over $\mathbb Q$, and to characteristic 0, parallel weight 1 Hilbert modular forms over $\mathbb Q(\sqrt{19 \cdot 151})$.
