Let $L$ be a finite abelian extension of $\mathbb{Q}$ and let $m$ be a positive integer such that $L\subset\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive $m$-th root of unity. Let $a$ be an integer coprime to $m$. Then the Artin symbol $(\frac{L}{a})$ is the automorphism of $L$ obtained by restricting to $L$ the automorphism $\phi$ of $\mathbb{Q}(\zeta)$ determined by $(\zeta\mapsto\zeta^a)$.
My question is, why is, $\phi(L)\subset L$?