norm and trace of algebraic number
Let be an algebraic number field and an element of . The norm and the trace of in the field extension both are rational numbers and especially rational integers in the case is an algebraic integer. If is another element of , then
1. The notions norm and trace were originally introduced in German as “die Norm” and “die Spur”. Therefore in German and many other literature the symbol of trace is S, Sp or sp. Nowadays the symbols T and Tr are common.
2. The norm and trace of an algebraic number in the field extension , i.e. the product and sum of all algebraic conjugates of , are called the absolute norm and the absolute trace of . Formulae like (1) concerning the absolute norms and traces are not sensible.
An algebraic integer is a unit if and only if
i.e. iff the absolute norm of is a rational unit. Thus in the minimal polynomial of an algebraic unit is always .
|Title||norm and trace of algebraic number|
|Date of creation||2013-03-22 15:19:08|
|Last modified on||2013-03-22 15:19:08|
|Last modified by||pahio (2872)|