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

Lemma.  Let ${\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_s$ 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$\mathbb{Q}\mathtt{}\mathrm{(}\mathrm{\vartheta }\mathrm{)}$-conjugates.  If$P(x_1,x_2,\mathrm{\dots },x_s)$ is a polynomial with rational coefficients and if

$$P({\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_s)=\mathrm{\hspace{0.33em}0},$$

then also

$$P({\alpha }_1^{(j)},{\alpha }_2^{(j)},\mathrm{\dots },{\alpha }_s^{(j)})=\mathrm{\hspace{0.33em}0}$$

for each  $j=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },n$.  In the special case of two elements $\alpha$ and $\beta$ of$\mathbb{Q}(\vartheta )$ one may infer the formulae

${(\alpha \beta )}^{(j)}={\alpha }^{(j)}{\beta }^{(j)},{(\alpha +\beta )}^{(j)}={\alpha }^{(j)}+{\beta }^{(j)}.$(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

$$\text{N}(\alpha \beta )=\text{N}(\alpha )\text{N}(\beta ),\text{S}(\alpha +\beta )=\text{S}(\alpha )+\text{S}(\beta ).$$

Cf. the entry norm and trace of algebraic number.