English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 someremarkable properties. Serre related the function on primes $N_p(f_n)$, for afixed $n \leq 4$ 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 tocharacteristic 0, parallel weight 1 Hilbert modular forms over $\mathbbQ(\sqrt{19 \cdot 151})$.

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