conjugate fields

If  ${\vartheta }_1,{\vartheta }_2,\mathrm{\dots },{\vartheta }_n$  are the algebraic conjugates of the algebraic number ${\vartheta }_1$, then the algebraic number fields  $\mathbb{Q}({\vartheta }_1),\mathbb{Q}({\vartheta }_2),\mathrm{\dots },\mathbb{Q}({\vartheta }_n)$ are the conjugate fields of $\mathbb{Q}({\vartheta }_1)$.

Notice that the conjugate fields of $\mathbb{Q}({\vartheta }_1)$ are always isomorphic but not necessarily distinct.

All conjugate fields are equal, i.e. (http://planetmath.org/Ie) $\mathbb{Q}({\vartheta }_1)=\mathbb{Q}({\vartheta }_2)=\mathrm{\dots }=\mathbb{Q}({\vartheta }_n)$, or equivalently ${\vartheta }_1,\mathrm{\dots },{\vartheta }_n$ belong to $\mathbb{Q}({\vartheta }_1)$, if and only if the extension $\mathbb{Q}({\vartheta }_1)/\mathbb{Q}$ is a Galois extension of fields. The reason for this is that if ${\vartheta }_1$ is an algebraic number and $m(x)$ is the minimal polynomial of ${\vartheta }_1$ then the roots of $m(x)$ are precisely the algebraic conjugates of ${\vartheta }_1$.

For example, let ${\vartheta }_1=\sqrt{2}$. Then its only conjugate is ${\vartheta }_2=-\sqrt{2}$ and $\mathbb{Q}(\sqrt{2})$ is Galois and contains both ${\vartheta }_1$ and ${\vartheta }_2$. Similarly, let $p$ be a prime and let ${\vartheta }_1=\zeta$ be a primitive $p$th root of unity (http://planetmath.org/PrimitiveRootOfUnity). Then the algebraic conjugates of $\zeta$ are ${\zeta }^2,\mathrm{\dots },{\zeta }^{p-1}$ and so all conjugate fields are equal to $\mathbb{Q}(\zeta )$ and the extension $\mathbb{Q}(\zeta )/\mathbb{Q}$ is Galois. It is a cyclotomic extension of $\mathbb{Q}$.

Now let ${\vartheta }_1=\sqrt[3]{2}$ and let $\zeta$ be a primitive $3$rd root of unity (i.e. $\zeta$ is a root of $x^2+x+1$, so we can pick $\zeta =\frac{-1+\sqrt{-3}}{2}$). Then the conjugates of ${\vartheta }_1$ are ${\vartheta }_1$, ${\vartheta }_2=\zeta \sqrt[3]{2}$, and ${\vartheta }_3={\zeta }^2\sqrt[3]{2}$. The three conjugate fields $\mathbb{Q}({\vartheta }_1)$, $\mathbb{Q}({\vartheta }_2)$, and $\mathbb{Q}({\vartheta }_3)$ are distinct in this case. The Galois closure of each of these fields is $\mathbb{Q}(\zeta ,\sqrt[3]{2})$.