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 $\alpha$.

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