# characteristic polynomial of algebraic number

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

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

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 $\alpha$.

Title characteristic polynomial of algebraic number CharacteristicPolynomialOfAlgebraicNumber 2013-03-22 19:08:41 2013-03-22 19:08:41 pahio (2872) pahio (2872) 14 pahio (2872) Definition msc 15A18 msc 12F05 msc 11R04 RationalIntegersInIdeals DegreeOfAlgebraicNumber characteristic polynomial characteristic equation $\mathbb{Q}(\vartheta)$-conjugates $K$-conjugates