# conjugate fields

If  $\vartheta_{1},\,\vartheta_{2},\,\ldots,\,\vartheta_{n}$  are the algebraic conjugates of the algebraic number $\vartheta_{1}$, then the algebraic number fields$\mathbb{Q}(\vartheta_{1}),\,\mathbb{Q}(\vartheta_{2}),\,\ldots,\,\mathbb{Q}(% \vartheta_{n})$  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) $\mathbb{Q}(\vartheta_{1})=\mathbb{Q}(\vartheta_{2})=\ldots=\mathbb{Q}(% \vartheta_{n})$, or equivalently $\vartheta_{1},\ldots,\vartheta_{n}$ 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 $\zeta^{2},\ldots,\zeta^{p-1}$ 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 $\mathbb{Q}(\zeta,\sqrt[3]{2})$.

Title conjugate fields ConjugateFields 2013-03-22 17:10:28 2013-03-22 17:10:28 pahio (2872) pahio (2872) 10 pahio (2872) Definition msc 12F05 msc 11R04 PropertiesOfMathbbQvarthetaConjugates