Is $\cos x$ irreducible as a power series? Let $\mathbb{Q}_{\mathrm{ent}}[[x]]$ be the ring of entire functions with rational coefficients. Is $$ \cos x \;=\; \sum_{n=0}^\infty (-1)^n\\!\frac{x^{2n}}{(2n)!} $$ irreducible in $\mathbb{Q}_\mathrm{ent}[[x]]$?

We have the trigonometric identity

$$\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} = \left(\cos \frac{x}{2} - \sin \frac{x}{2}\right)\left(\cos\frac{x}{2} + \sin \frac{x}{2}\right),$$

and both factors are entire power series with rational coefficients, and not units, since they have zeros in $\mathbb{C}$.