Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients such that 0 is not a root. Show that there exists a unique power series $f \in \mathbf{Q}\llbracket x \rrbracket$ satisfying $Q \cdot f = 1$.
Show that if $Q$ has integer coefficients and its constant term $c_0$ equals 1 or $-1$, then this unique power series $f$ has integer coefficients.