# independence of characteristic polynomial on primitive element

The simple field extension $\mathbb{Q}(\vartheta)/\mathbb{Q}$ where $\vartheta$ is an algebraic number of degree (http://planetmath.org/DegreeOfAnAlgebraicNumber) $n$ may be determined also by using another primitive element $\eta$.  Then we have

 $\eta\in\mathbb{Q}(\vartheta),$

whence, by the entry degree of algebraic number, the degree of $\eta$ divides the degree of $\vartheta$.  But also

 $\vartheta\in\mathbb{Q}(\eta),$

whence the degree of $\vartheta$ divides the degree of $\eta$.  Therefore any possible primitive element of the field extension has the same degree $n$.  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

 $g(x)\;:=\;\prod_{i=1}^{n}[x-r(\vartheta_{i})]\;=\;\prod_{i=1}^{n}(x-\alpha^{(i% )})$

of an element $\alpha$ of the algebraic number field $\mathbb{Q}(\vartheta)$ is based on the primitive element $\vartheta$, the equation

 $\displaystyle g(x)\;=\;(x-\alpha_{1})^{m}(x-\alpha_{2})^{m}\cdots(x-\alpha_{k}% )^{m}$ (1)

in the entry http://planetmath.org/node/12050degree of algebraic number shows that the polynomial is fully determined by the algebraic conjugates of $\alpha$ itself and the number $m$ which equals the degree $n$ divided by the degree $k$ of $\alpha$.

The above stated makes meaningful to define the norm and the trace functions in an algebraic number field as follows.

Definition.  If $\alpha$ is an element of the number field $\mathbb{Q}(\vartheta)$, then the norm $\mbox{N}(\alpha)$ and the trace $\mbox{S}(\alpha)$ of $\alpha$ are the product and the sum, respectively, of all http://planetmath.org/node/12046$\mathbb{Q}(\vartheta)$-conjugates $\alpha^{(i)}$ of $\alpha$.

Since the coefficients of the characteristic equation of $\alpha$ are rational, one has

 $\mbox{N}\!:\,\mathbb{Q}(\vartheta)\to\mathbb{Q}\quad\mbox{and}\quad\mbox{S}\!:% \,\mathbb{Q}(\vartheta)\to\mathbb{Q}.$

In fact, one can infer from (1) that

 $\displaystyle\mbox{N}(\alpha)\;=\;a_{k}^{m},\qquad\mbox{S}(\alpha)\;=\;-ma_{1},$ (2)

where $x^{k}\!+\!a_{1}x^{k-1}\!+\ldots+\!a_{k}$ is the minimal polynomial of $\alpha$.

 Title independence of characteristic polynomial on primitive element Canonical name IndependenceOfCharacteristicPolynomialOnPrimitiveElement Date of creation 2014-02-04 8:07:18 Last modified on 2014-02-04 8:07:18 Owner pahio (2872) Last modified by pahio (2872) Numerical id 10 Author pahio (2872) Entry type Topic Classification msc 11R04 Classification msc 12F05 Classification msc 11C08 Classification msc 12E05 Synonym norm and trace functions in number field Related topic Norm Related topic NormAndTraceOfAlgebraicNumber Related topic PropertiesOfMathbbQvarthetaConjugates Related topic DiscriminantInAlgebraicNumberField Defines norm in number field Defines trace in number field Defines norm Defines trace