It can be observed that a matrix of order $5$ over $\mathbb{R}$ has at least one eigenvalue in $\mathbb{R}$.
What if we consider a finite field?
For example, over $\mathbb{Z}_2$, a matrix having characteristic polynomial $x^4(x-1)+1$ cannot have an eigenvalue from $\mathbb{Z}_2$.
Can such a matrix exist?
Given any polynomial $f$ over a field $F$, you can always construct a companion matrix over $F$ whose characteristic polynomial and minimal polynomial are both equal to $f$. (Other properties of $f$, such as whether $f$ has a zero over $F$, or whether $f$ is irreducible over $F$, are irrelevant.) In your case, $x^4(x-1)+1=x^5-x^4+1$ is the characteristic polynomial of the matrix $$ C=\pmatrix{ 0&0&0&0&-1\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&1}. $$