Algebraic number field basis

Hi! I tried to show the following but i failed: We have a finite galois extension $\displaystyle L/\mathbb{Q}$ with commutative galois group and $\displaystyle O_L$ the ring of algebraic integers in L. Suppose there exists $\displaystyle x \in O_L$ so that the set $\displaystyle \{\sigma(x): \sigma \in $ Gal($\displaystyle L/\mathbb{Q})\}$ forms a $\displaystyle \mathbb{Z}$-Basis of $\displaystyle O_L$.

Then for every field $\displaystyle K \subset L$ there also exists $\displaystyle y \in O_K$ so that $\displaystyle \{\phi(y): \phi \in $ Gal($\displaystyle K/\mathbb{Q})\}$ forms a $\displaystyle \mathbb{Z}$-Basis of $\displaystyle O_K$.

I could show that $\displaystyle K/\mathbb{Q}$ is normal and hence each $\displaystyle \phi \in $ Gal$\displaystyle (K/\mathbb{Q})$ is just a restriction of $\displaystyle \sigma \in $ Gal $\displaystyle (L/\mathbb{Q})$. But i dont know how to continue the argument. Can anybody please help me?

Greetings

Banach