Factorization in $\mathbb{Z}[\sqrt[4]{3}]$

Question

Let $p$ a prime number such that $p≡3$ (m0d $4$), and $p>3$, discuss the factorization of $(p)$ in $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt[4]{3})$.

I know that the ring of integers in this case is $\mathbb{Z}[\sqrt[4]{3}]$,and by Kummer's theorem, one can decide the decomposition based on factorization of $f(x)=x^4−3$ (mod $p$).

There are five possible cases:

1. $f(x)$ factors into $4$ linear factors in $\mathbb{F}_{p}[x]$.
2. $f(x)$ factors into $2$ linear factors and $1$ irreducible quadratic factors in $\mathbb{F}_{p}[x]$.
3. $f(x)$ factors into $2 $ irreducible quadratic factors in $\mathbb{F}_{p}[x]$.
4. $f(x)$ factors into $1 $ linear factor and $1$ irreducible cubic factor in $\mathbb{F}_{p}[x]$.
5. $f(x)$ is irreducible in $\mathbb{F}_{p}[x]$.

I don't know how to go further than this!