Polynomial ring factorization

Show that

Let $f, g$ be monic polynomials from $\mathbb {Z}{}[X]$. Show that in the expression $\gcd (f, g)=f u+g v$ with $u, v \in \mathbb {Z}{}[X]$ one may take $\operatorname {deg} u<\operatorname {deg} g$, $\operatorname {deg} v<\operatorname {deg} f$.

Are the rings $\mathbb {Z}{}[\sqrt{-3}]$ and $\mathbb {Z}_{8}[X]$ factorial?

Factor into irreducible factors in $\mathbb {Z}{}[X]$ the polynomials $X^{n}-1$ for $5 \leqslant n \leqslant 12$.

Prove that the irreducible factors of the homogeneous polynomial

are homogeneous, and $f(X, Y)$ is irreducible if and only if the polynomial $f(X, 1)=a_{0} X^{n}+a_{1} X^{n-1}+\ldots +a_{n-1} X+a_{n} \in \mathbb {Q}{}[X]$ is irreducible.

Let $P$ be a field and $f(X)=\sum_{i \geqslant 0} a_{i} X^{i}$ a formal power series from $P[[X]]$ (see Exercise 5 from § 2). The condition $a_{0} \neq 0$, or, equivalently, $\omega (f)=0$, is necessary and sufficient for the existence of a power series $g(X) \in P[[X]]$ inverse to $f: f g=1$. For example, $(1-X)^{-1}=\sum_{i \geqslant 0} X^{i}$. Up to associates, $X$ is the unique prime element in $P[[X]]$. The ring $P[[X]]$ is factorial. Justify these statements.

Show that $\operatorname {det}\left(x_{i j}\right)=\sum_{\pi \in S_{n}} \varepsilon_{\pi } x_{\pi (1), 1} \ldots x_{\pi (n), n}$ is an irreducible homogeneous polynomial of degree $n$ in $n^{2}$ independent variables $x_{i j}$.