I have a finite Galois extension of number fields $L/K$ with group $G$. Let $\mathcal O_L$ and $\mathcal O_K$ be the respective rings of algebraic integers. I want to show that $\mathcal O_L$ is isomorphic to a $\mathcal O_K$-lattice in $K[G]$.
By the normal basis theorem I know that $L$ is equal to $K[G].\alpha$ where $\alpha \in L$ hence $\varphi:L\cong K[G]$ as $K[G]$-modules. I want to show that $\varphi(\mathcal O_L)$ is the desired lattice.
Since $\mathcal O_L$ is a finitely generated free $\mathcal O_K$-submodule of $L$, I can think of $\varphi(\mathcal O_L)$ as a f.g. free $\mathcal O_K$-submodule of $K[G]$. (How do I make this rigorous?)
To show that $K.\varphi(\mathcal O_L)=K[G]$, I choose $x\in K[G]$. Then $x=\varphi(y)$ for a unique $y\in L$. I can choose $b\in \mathcal O_K$ s.t. $by=a\in \mathcal O_L.$ Then $bx=b\varphi(y)=\varphi(by)=\varphi(a)\in \varphi(\mathcal O_L)$. So $x\in K.\varphi(\mathcal O_L).$
Am I on the right track?