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

Question

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.

Any advice?

Answer 1

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.