Let $K/F$ be an abelian extension (a Galois extension $K/F$ such that $Gal(K/F)$ is abelian) of degree at least $2$.
Prove that there exists a tower of fields $F=K_{0} \subset K_1 \subset \cdots \subset K_r=K$ such that $K_{i+1}/{K_i}$ is a Galois extension of prime degree (i.e. $[K_{i+1}:K_i]=p$, where $p$ is prime) for each $i$.
How do I prove this claim? Any help will be appreciated. Thanks.