characteristic polynomial of algebraic number
Let be an element of the number field and
the canonical polynomial of with respect to . We consider the numbers
and form the equation
the roots of which are the numbers (1) and only these. The coefficients of the polynomial are symmetric polynomials in the numbers and also symmetric polynomials in the numbers . The fundamental theorem of symmetric polynomials implies now that the symmetric polynomials in the roots of the equation belong to the ring determined by the coefficients of the equation and of the canonical polynomial ; thus the numbers are rational (whence the degree of is at most equal to ).
It is not hard to show (see the entry degree of algebraic number) of that the degree of divides and that the numbers (1) consist of and its algebraic conjugates , each of which appears in (1) exactly times. In fact, where is the minimal polynomial of (consequently, the coefficients
are integers if is an algebraic integer).
The polynomial is the characteristic polynomial (in German Hauptpolynom) of the element of the algebraic number field and the equation the characteristic equation (Hauptgleichung) of . See the independence of characteristic polynomial on primitive element.
So, the roots of the characteristic equation of are . They are called the -conjugates of ; they all are algebraic conjugates of .
|Title||characteristic polynomial of algebraic number|
|Date of creation||2013-03-22 19:08:41|
|Last modified on||2013-03-22 19:08:41|
|Last modified by||pahio (2872)|