When it comes to polynomials whose zeros are $n^{th}$ roots of irrational quadratic integers (mainly units), like the above examples, what is the general method in determining the degree of each irreducible factor in $\mathbb{F}_{p}[X]$? All outside a list of finitely many primes $p$ of course. I chose these two specific ones for two reasons;
1). The Galois group of the polynomials $f(X)=X^4-X^2-1$ and $g(X)=X^4-2X^2-1$ are not abelian, unlike the $h(X)=X^4-4X^2+1$ whose roots are
$$\pm\sqrt{2\pm\sqrt{3}}=\pm\frac{1\pm\sqrt{3}}{\sqrt{2}}$$
alluding to quadratic reciprocity. And adjoining a root of $f$ or $g$ to $\mathbb{Q}$ isn't of the form $\mathbb{Q}(\sqrt[4]{n})$.
2) I had tried doing so without success and forgot until I came back, after learning some reciprocity, and was able to remember on the off chance a way to do so for these two specific cases.
For an example computation, the zeros of $f(X)$ are
$$\pm\sqrt{\frac{1+\sqrt{5}}{2}}, \pm\sqrt{\frac{1-\sqrt{5}}{2}}.$$
Determining the action of the map $T_{p}: r \mapsto r^{p} \pmod{p}$ for $$r \in \left\{\sqrt{\frac{1+\sqrt{5}}{2}}, -\sqrt{\frac{1+\sqrt{5}}{2}}, \sqrt{\frac{1-\sqrt{5}}{2}}, -\sqrt{\frac{1+\sqrt{5}}{2}}\right\}$$ an ordered set can be worked with by knowing properties of the quartic residue symbol $$\Big[\frac{5}{p}\Big]_{4}=5^{\frac{p-1}{4}} \pmod{p}$$ and that $$\sin\frac{2\pi}{5}=\sqrt{\frac{5+\sqrt{5}}{8}}=\frac{\sqrt[4]{5}}{2}\cdot\sqrt{\frac{1+\sqrt{5}}{2}}.$$
From there, it becomes a lot easier to factor in because the action of $T_{p}$ is straightforward with the minimal polynomial of $\sin\frac{2\pi}{5}$, and the action is relatively accessible with the residue symbol.
The same goes for determining the action of $T_{p}$ for $g(X)$ by considering quartic residue symbol $\left[\dfrac{2}{p}\right]_{4}$ and $\cos\dfrac{\pi}{8}=\dfrac{\sqrt{2+\sqrt{2}}}{2}$.
But this approach feels too ad hoc, and doesn't say anything about how to factor anything like, for example, $\displaystyle f(X^n)$ in general. What kind of tools are used to find how $h(X^{3})$ in $\mathbb{F}_{p}[X]$? $h(X^{\frac{3}{2}})$?
Also, phrased using the language of fields, this raises the question of when the field $\mathbb{Q}(\sqrt{q})$ is Galois over $\mathbb{Q}$ for $q$ an irrational square-free quadratic integer
Recall that any irreducible polynomial of degree $n$ factors into linear factors in $\Bbb F_{p^n}$. As such you can compute $\gcd(p(x), x^{p^k}-x)$ for all highest-prime-power $k\le n$. We do this for all of them as your polynomial may not be irreducible (so it may be a product of irreducible factors, but we know all such factors have degree at most $n$). So we compute for $k=q^j$ for prime powers $q^j\le n$ and so that $j$ is maximal (since $j<j'\implies q^{j'}|q^j$). Then this gives you all possible common factors with each of the $x^{p^{q^j}}-x$ which generate all possible roots for $p(x)$. This can be automated by programming a computer to run the Euclidean algorithm on each of the polynomials and spitting out the result.