characteristic polynomial of algebraic number

Let $\vartheta$ be an algebraic number of degree $n$, $f(x)$ its minimal polynomial and

$${\vartheta }_1=\vartheta ,{\vartheta }_2,\mathrm{\dots },{\vartheta }_n$$

its algebraic conjugates.

Let $\alpha$ be an element of the number field $\mathbb{Q}(\vartheta )$ and

$$r(x):=c_0+c_{1}x+\mathrm{\dots }+c_{n-1}{x}^{n-1}$$

the canonical polynomial of $\alpha$ with respect to $\vartheta$.  We consider the numbers

$r({\vartheta }_1)=\alpha :={\alpha }^{(1)},r({\vartheta }_2):={\alpha }^{(2)},\mathrm{\dots },r({\vartheta }_n):={\alpha }^{(n)}$(1)

and form the equation

$$g(x):=\prod _{i=1}^n[x-r({\vartheta }_i)]=(x-{\alpha }^{(1)})(x-{\alpha }^{(2)})\mathrm{\cdots }(x-{\alpha }^{(n)})=x^n+g_1{x}^{n-1}+\mathrm{\dots }+g_n=\mathrm{\hspace{0.33em}0},$$

the roots of which are the numbers (1) and only these.  The coefficients $g_i$ of the polynomial $g(x)$ are symmetric polynomials in the numbers ${\vartheta }_1,{\vartheta }_2,\mathrm{\dots },{\vartheta }_n$ and also symmetric polynomials in the numbers ${\alpha }^{(i)}$.  The fundamental theorem of symmetric polynomials implies now that the symmetric polynomials $g_i$ in the roots ${\vartheta }_i$ of the equation  $f(x)=0$  belong to the ring determined by the coefficients of the equation and of the canonical polynomial $r(x)$; thus the numbers $g_i$ are rational (whence the degree of $\alpha$ is at most equal to $n$).

It is not hard to show (see the entry degree of algebraic number) of that the degree $k$ of $\alpha$ divides $n$ and that the numbers (1) consist of $\alpha$ and its algebraic conjugates ${\alpha }_2,\mathrm{\dots },{\alpha }_k$, each of which appears in (1) exactly  $\frac{n}{k}=m$  times.  In fact,  $g(x)={[a(x)]}^m$  where $a(x)$ is the minimal polynomial of $\alpha$ (consequently, the coefficients$g_i$ are integers if $\alpha$ is an algebraic integer).

The polynomial $g(x)$ is the characteristic polynomial (in German Hauptpolynom) of the element $\alpha$ of the algebraic number field $\mathbb{Q}(\vartheta )$ and the equation  $g(x)=0$ the characteristic equation (Hauptgleichung) of $\alpha$.  See the independence of characteristic polynomial on primitive element.

So, the roots of the characteristic equation of $\alpha$ are ${\alpha }^{(1)},{\alpha }^{(2)},\mathrm{\dots },{\alpha }^{(n)}$.  They are called the $\mathbb{Q}(\vartheta )$-conjugates of $\alpha$; they all are algebraic conjugates of$\alpha$.