21. Let $\alpha$ be an algebraic number. Define the set of polynomials
$$P(\alpha)=\{f(x): f(x) \in \mathbb{Q}[x], f(\alpha)=0\}.$$
(i) Prove: For non-zero polynomials in $P(\alpha)$, the following three properties of $h(x)$ are equivalent: (a) $h(x)$ is the polynomial of lowest degree in $P(\alpha)$; (b) $f(x) \in P(\alpha)$ if and only if $h(x)$ divides $f(x)$ in $\mathbb{Q}[x]$; (c) $h(x)$ is an irreducible polynomial in $\mathbb{Q}[x]$ that belongs to $P(\alpha)$.
(ii) Prove: There exists a unique polynomial $g(x)=g(x ; \alpha)$ in $P(\alpha)$ with leading coefficient 1 that has the properties (a), (b), and (c) from (i). We call $g(x)$ the minimal polynomial of the algebraic number $\alpha$, and the degree of $g(x)$ is called the degree of the algebraic number $\alpha$.
(iii) Prove: $\alpha$ is a linear algebraic number if and only if $\alpha \in \mathbb{Q}$; $\alpha$ is a linear algebraic integer if and only if $\alpha \in \mathbb{Z}$.
(iv) Prove: $\alpha$ is an algebraic integer if and only if its minimal polynomial $g(x ; \alpha) \in \mathbb{Z}[x]$.
(v) An algebraic number $\alpha$ is called a unit if both $\alpha$ and $\alpha^{-1}$ are algebraic integers. Prove: $\alpha$ is a unit if and only if its minimal polynomial $g(x ; \alpha) \in \mathbb{Z}[x]$, and its constant term is $\pm 1$.