I. Cubics
The cubic $$x^3+x^2-2x-1=0$$ and its solution in terms of $\cos(n\pi/7)$ is well-known. A general formula for other primes $\color{blue}{p=6m+1}$ is discussed in this post.
II. Quintics
We have $$x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0,\quad\quad x =\sum_{k=1}^{2}\,\exp\Bigl(\tfrac{2\pi\, i\, (10^k)}{11}\Bigr)\\ x^5 + x^4 - 12 x^3 - 21 x^2 + x + 5=0,\quad\quad x =\sum_{k=1}^{6}\,\exp\Bigl(\tfrac{2\pi\, i\, (6^k)}{31}\Bigr)$$
and so on as in this post. The general form for prime $\color{blue}{p=10m+1}$ is $$x^5 + x^4 - 4m x^3 +a x^2 + bx + c=0$$ for some integer $a,b,c$
Q: Can we express $a,b,c$ as a polynomial in $p$ and some specified Diophantine equation $F(u,v)$ just like in the cubic case?
