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

$g(x)={(x-{\alpha }_1)}^m{(x-{\alpha }_2)}^m\mathrm{\cdots }{(x-{\alpha }_k)}^m$(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$\text{N}(\alpha )$ and the trace $\text{S}(\alpha )$ of $\alpha$ are the product and the sum, respectively, of all http://planetmath.org/node/12046$\mathbb{Q}\mathtt{}\mathrm{(}\mathrm{\vartheta }\mathrm{)}$-conjugates ${\alpha }^{(i)}$ of $\alpha$.

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

$$\text{N}:\mathbb{Q}(\vartheta )\to \mathbb{Q}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\text{S}:\mathbb{Q}(\vartheta )\to \mathbb{Q}.$$

In fact, one can infer from (1) that

$\text{N}(\alpha )=a_k^m,\text{S}(\alpha )=-m{a}_1,$(2)

where $x^k+a_1{x}^{k-1}+\mathrm{\dots }+a_k$ is the minimal polynomial of $\alpha$.