There is a simple criteria for detecting when $p(x)=x^4-7x^2-3x+1$ split completely mod a prime $p$. 

Let
$$a_1=c_1=d_1=0, b_1=1$$ and set
$$a_{n+1}=-d_n,$$
$$b_{n+1}=a_n+3d_n,$$
$$c_{n+1}=b_n+7d_n,$$
$$d_{n+1}=c_n,$$ 
$$A_n=gcd(a_n-1,b_n,c_n,d_n),$$

then the polynomial $p(x)$ split completely mod $p$ if and only if $p|A_{p-1}$. 

The recursion comes from the orbit of iterating the companion matrix of $p(x)$ and then taking a GCD. It is rather unsophisticated but it works for all algebraic integer of any degree which is not a root of unity.