I am currently working on this problem:
Let $$P(x) = x^4+x^3+x^2+x+1$$ show that there is an extension $K$ of $\mathbb Q$ with $[ K : \mathbb Q ] = 4$ over which $P(x)$ splits into linear factors.
I know I just need to find the roots in $\mathbb C$ or $\mathbb R$ and adjoin them to $\mathbb Q$ to create the field over which it can split. I just can't seem to find any roots without using a calculator, and that just leaves me with decimal solutions. It seems if I had a nice solution in radicals it would be easy to find the necessary extensions. So I was wondering if this polynomial had simple solutions in radicals. Also, should I know how to solve equations like this without a calculator?