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# characteristic polynomial of algebraic number

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

$\vartheta_{1}=\vartheta,\;\vartheta_{2},\;\ldots,\;\vartheta_{n}$ |

its algebraic conjugates.

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

$r(x)\;:=\;c_{0}+c_{1}x+\ldots+c_{{n-1}}x^{{n-1}}$ |

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

$\displaystyle r(\vartheta_{1})\;=\;\alpha\;:=\;\alpha^{{(1)}},\quad r(% \vartheta_{2})\;:=\;\alpha^{{(2)}},\quad\ldots,\quad r(\vartheta_{n})\;:=\;% \alpha^{{(n)}}$ | (1) |

and form the equation

$g(x)\;:=\;\prod_{{i=1}}^{n}[x\!-\!r(\vartheta_{i})]\;=\;(x\!-\!\alpha^{{(1)}})% (x\!-\!\alpha^{{(2)}})\cdots(x\!-\!\alpha^{{(n)}})\;=\;x^{n}\!+\!g_{1}x^{{n-1}% }\!+\!\ldots\!+\!g_{n}\;=\;0,$ |

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 $\vartheta_{1},\,\vartheta_{2},\,\ldots,\,\vartheta_{n}$ 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 $\alpha_{2},\,\ldots,\,\alpha_{k}$, 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 $\alpha^{{(1)}},\,\alpha^{{(2)}},\,\ldots,\,\alpha^{{(n)}}$. They are called the $\mathbb{Q}(\vartheta)$*-conjugates* of $\alpha$; they all are algebraic conjugates of
$\alpha$.

## Mathematics Subject Classification

15A18*no label found*12F05

*no label found*11R04

*no label found*

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