I have to prove that any non-zero element $b(z)\in\mathbb{C}((z))$ (the field of fractions of the ring $\mathbb{C}[[z]]$ of formal power series) has a unique presentation of the form $b(z)=z^ma(z)$ with $a(z)\in\mathbb{C}[[z]]$, $a_0\neq 0$ and $m\in\mathbb{Z}$.
I know that the ideals in the ring are of the form $(z^m)$ and that any formal power series with $a_0\neq 0$ is a unit. Can I somehow use this to prove the statement or how else should I prove it?
$b(z)=f(z)/g(z)$ with $f,g\in\mathbb C[[z]]$, $g\ne0$. Write $f(z)=z^kf_1(z)$ with $f_1(0)\ne0$, and $g(z)=z^lg_1(z)$ with $g_1(0)\ne0$. Then $b(z)=z^{k-l}f_1(z)(g_1(z))^{-1}$. Now set $a(z)=f_1(z)(g_1(z))^{-1}$ and notice that $a(0)\ne0$.
The uniqueness is left as an exercise. (You can ask for help if need it.)