# For $m$ distinct fields among $\mathbb{Q}(\theta_1),\ldots,\mathbb{Q}(\theta_n)$ prove that $m\mid n$ and each field occurs $n/m$ times

I'm having some trouble with this problem, and I wanted to know if someone could help me out. Let $K=\mathbb{Q}(\theta)$ be an algebraic number field of degree $n$. Let $\theta_1=\theta,\ldots,\theta_n$ be the conjugates of $\theta$ over $\mathbb{Q}$. Suppose there are exactly $m$ distinct fields among $\mathbb{Q}(\theta_1),\ldots,\mathbb{Q}(\theta_n)$ prove that $m\mid n$ and each field occurs $n/m$ times.

• Well, a simple example is $\mathbb Q(i)$. $i$'s conjugates are $\pm i$, but there is only one distinct field extension. Dec 8, 2014 at 20:56
• $\mathbb Q(\sqrt[3]{2})$? Dec 8, 2014 at 21:26
Say $S$ is the set of $\theta_i$'s that are in $\mathbb Q(\theta_1)$. Let $\sigma_j$ be an automorphism that sends $\theta_1\mapsto\theta_j$. Then the set $\sigma_j(S)$ is exactly the set of $\theta_i$'s in $\mathbb Q(\theta_j)$. Thus, each of the distinct fields among them has the same number of $\theta_i$'s, which means that the $n$ of them are partitioned into $m$ same-size sets of $n/m$ elements.