independence of characteristic polynomial on primitive element
whence, by the entry degree of algebraic number, the degree of divides the degree of . But also
whence the degree of divides the degree of . Therefore any possible primitive element of the field extension has the same degree . This number is the degree of the number field (http://planetmath.org/NumberField), i.e. the degree of the field extension, as comes clear from the entry canonical form of element of number field.
Although the characteristic polynomial
of an element of the algebraic number field is based on the primitive element , the equation
in the entry http://planetmath.org/node/12050degree of algebraic number shows that the polynomial is fully determined by the algebraic conjugates of itself and the number which equals the degree divided by the degree of
The above stated makes meaningful to define the norm and the trace functions in an algebraic number field as follows.
In fact, one can infer from (1) that
where is the minimal polynomial of .
|Title||independence of characteristic polynomial on primitive element|
|Date of creation||2014-02-04 8:07:18|
|Last modified on||2014-02-04 8:07:18|
|Last modified by||pahio (2872)|
|Synonym||norm and trace functions in number field|
|Defines||norm in number field|
|Defines||trace in number field|