# properties of $\mathbb{Q}(\vartheta)$-conjugates

Lemma.  Let $\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{s}$ be algebraic numbers belonging to the number field $\mathbb{Q}(\vartheta)$ of degree (http://planetmath.org/NumberField) $n$ and $\alpha_{i}^{(j)}$ their http://planetmath.org/node/12046$\mathbb{Q}(\vartheta)$-conjugates.  If $P(x_{1},\,x_{2},\,\ldots,\,x_{s})$ is a polynomial with rational coefficients and if

 $P(\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{s})\;=\;0,$

then also

 $P(\alpha_{1}^{(j)},\,\alpha_{2}^{(j)},\,\ldots,\,\alpha_{s}^{(j)})\;=\;0$

for each  $j=1,\,2,\,\ldots,\,n$.  In the special case of two elements $\alpha$ and $\beta$ of $\mathbb{Q}(\vartheta)$ one may infer the formulae

 $\displaystyle(\alpha\beta)^{(j)}\;=\;\alpha^{(j)}\beta^{(j)},\quad(\alpha\!+\!% \beta)^{(j)}\;=\;\alpha^{(j)}\!+\!\beta^{(j)}.$ (1)

The lemma implies easily the following theorems.

Theorem 1.  All conjugate fields of $\mathbb{Q}(\vartheta)$ are isomorphic.

Theorem 2.  The norm and the trace in the field $\mathbb{Q}(\vartheta)$ satisfy

 $\mbox{N}(\alpha\beta)\;=\;\mbox{N}(\alpha)\mbox{N}(\beta),\quad\mbox{S}(\alpha% \!+\!\beta)\;=\;\mbox{S}(\alpha)\!+\!\mbox{S}(\beta).$
Title properties of $\mathbb{Q}(\vartheta)$-conjugates PropertiesOfmathbbQvarthetaconjugates 2013-03-22 19:09:14 2013-03-22 19:09:14 pahio (2872) pahio (2872) 7 pahio (2872) Topic msc 11R04 msc 11C08 msc 12E05 msc 12F05 ConjugateFields IndependenceOfCharacteristicPolynomialOnPrimitiveElement