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 $\alpha$.

Title	characteristic polynomial of algebraic number
Canonical name	CharacteristicPolynomialOfAlgebraicNumber
Date of creation	2013-03-22 19:08:41
Last modified on	2013-03-22 19:08:41
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	14
Author	pahio (2872)
Entry type	Definition
Classification	msc 15A18
Classification	msc 12F05
Classification	msc 11R04
Related topic	RationalIntegersInIdeals
Related topic	DegreeOfAlgebraicNumber
Defines	characteristic polynomial
Defines	characteristic equation
Defines	$\mathbb{Q}(\vartheta )$-conjugates
Defines	$K$-conjugates