In this question about finding the Galois group of the polynomial $p(x) = x^4+2utx^2+ t$ over the field $\mathbb{C}(t)$ is raised the question of whether $\sqrt t$ belongs to $\mathbb{C}(t)[\alpha]$ where $\alpha$ is a root of $p$ in an algebraic extension.
It turns out that answering that question is equivalent to deciding whether the system of algebraic equations
$$\begin{cases} p_{0}^{2} - 2p_{1} p_{3} t - p_{2}^{2} t + 2 p_{3}^{2} t^{2} u - t &= 0\\ p_{0} p_{1} - p_{2} p_{3} t &= 0\\ 2p_{0} p_{2} + p_{1}^{2} - 4 p_{1} p_{3} t u - 2 p_{2}^{2} t u + 4 p_{3}^{2} t^{2} u^{2} - p_{3}^{2} t &= 0\\ p_{0} p_{3} + p_{1} p_{2} - 2 p_{2} p_{3} t u &= 0 \end{cases}$$
have a solution in variables $p_0, p_1, p_2, p_2 \in \mathbb{C}(t)$.
I tried a bit to manipulate those equations to find a contradiction, mainly regarding the degrees of the polynomials that can be used to write the $p_i$. With no success so far.
Hence my questions:
- What are the ways to tackle such a problem?
- Is there some programming package that can be used regarding the existence of solutions to a system of algebraic equations?
