Degree of compositum of two abelian extensions

Question

Let $L$ and $K$ be abelian extensions of $\mathbb{Q}$. Let $[L:\mathbb{Q}]=p^m$ and $[K:\mathbb{Q}]=p^n$, where $p\in\mathbb{Z}$ is a prime. Is it true that $[KL:\mathbb{Q}]$ is a power of $p$ ?

Answer 1

I found the solution. The answer is yes. We have

$$\text{Gal}(KL|\mathbb{Q})\hookrightarrow\text{Gal}(K|\mathbb{Q})\times\text{Gal}(L|\mathbb{Q})$$

As

$$|\text{Gal}(K|\mathbb{Q})\times\text{Gal}(L|\mathbb{Q})|=|\text{Gal}(K|\mathbb{Q})|\cdot|\text{Gal}(L|\mathbb{Q})|=p^{n+m}$$

$|\text{Gal} (KL|\mathbb{Q})|$ is also a power of $p$.