TITLE: Existence of homomorphism $\phi:\mathbb{Z}[[X]]\to\mathbb{Q}$ such that $\phi(X)=3/4$?
QUESTION [5 upvotes]: I'm curious as to whether there is a ring homomorphism from $\mathbb{Z} [[X]]$,the ring of formal power series in indeterminate $X$, to the rationals such that for every polynomial $p(X)$ in the ring of power series, $\phi(p(X))=p(3/4)$.

REPLY [2 votes]: The generating function $F \in \mathbb{Z}[[X]]$ of Catalan numbers $(C_n)_{n=0}^\infty$ is given by
$$F(X) = \sum_{n=0}^\infty C_nX^n = \frac{1-\sqrt{1-4X}}{2X}$$
and hence it satisfies
$$(1-2XF(X))^2 = 1-4X.$$
If such a homomorphism $\phi$ would exist, we would have
$$(1-2\phi(X)\phi(F(X)))^2 = 1-4\phi(X) \implies \left(\underbrace{1-\frac32\phi(F(X))}_{\in\mathbb{Q}}\right)^2 = 1-4\cdot \frac34 = -2$$
which is a contradiction.