Let $K$ be a number field, and let $P(t,X)$ be a monic polynomial in $X$ with coefficients in $K(t)$.

I would like to understand the set $T$ consisting of those $t_0 \in K$ such that the polynomial $P(t_0,X)$ is (totally) split over $K$.

**Q1.** Is there an algorithm to decide whether $T$ is empty, finite or infinite?

**Q2.** In the case $T$ is infinite, does there exist a non-constant rational function $f \in K(u)$ such that $t_0 \in T$ if and only if $t_0 = f(u_0)$ for some $u_0 \in K$?

**Q3.** In the case $T$ is infinite, is there an algorithm to find a non-constant rational function $f \in K(u)$ such that $P(f(u),X)$ splits over $K(u)$?

Here are some easy examples.

Example 1. Take $K=\mathbf{Q}$ and $P=X^2-t$. Obviously $T$ is the set of rational squares, and $f(u)=u^2$ satisfies the conditions in Q2 and Q3.

Example 2. Take $K$ arbitrary and $P=X^3-t$. If $t_0 \in K^\times$ and $P(t_0,X)$ splits over $K$, then $K$ must contain the cube roots of unity, and in this case the rational function $f(u)=u^3$ satisfies the conditions in Q2 and Q3.

My motivation for this question is that I'm trying to determine explicit equations for universal elliptic curves. For example, let $E_1(4)$ (resp. $E(4)$) be the universal elliptic curve over the modular curve $Y_1(4)$ (resp. $Y(4)$). A Weierstrass equation for $E_1(4)$ is given by $y^2+xy+ty=x^3+tx^2$ where $t$ is a generator of the function field of $Y_1(4)$ and the universal point of order 4 is given by $(0,0)$. Now determining $E(4)$ amounts to find those $t$ such that the $4$-division polynomial of $E_1(4)$ splits completely over $\mathbf{Q}(i)$.

So here is a concrete question: take $K=\mathbf{Q}(i)$ and $P(t,X)=X^4 + \frac12 X^3 + \frac32 tX^2 + 2t^2 X + t^3$. Can you find an explicit rational function $f \in K(u)$ such that $P(f(u),X)$ splits over $K(u)$?