We are given the formal power series $$ \alpha(x) = \sum_{k=0}^{\infty}a_k*x^k $$ and let us set deg($\alpha$) as the smallest $k = 0,1,2,..$ such that $a_0=a_1=...=a_{k-1}=0, a_k \not=0$.
I want to prove that $\alpha$ has an inverse in the set of formal power series with coefficients in $\mathbb{Q}$ if and only if deg($\alpha$) is zero.
We can use the Lemma that deg($\alpha\beta$) = deg($\alpha$) + deg($\beta$). The direction '$\Leftarrow$' is already understood. Proving '$\Rightarrow$': Let $\beta$ be the inverse of $\alpha$
$$deg(\alpha \beta) = deg(\alpha) + deg(\beta) = deg(1) = 0 $$ I don't know how to follow from this that $deg(\alpha) = 0$. Any help is much appreciated. Thank you.