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The simple field extension $\mathbb{Q}(\vartheta)/\mathbb{Q}$ where $\vartheta$ is an algebraic number of degree $n$ 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, 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 degree 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 $\mathbb{Q}(\vartheta)$-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 $\alpha$.

## Mathematics Subject Classification

11R04*no label found*12F05

*no label found*11C08

*no label found*12E05

*no label found*

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