Find the splitting field of $x^2 - \pi^4$ over $Q(\pi^4)$

Question

This is the first time I encountered a problem asking to find the splitting field of a polynomial with transcendental coefficients over $Q$ adjoined to a transcendental number over $Q$. I have no idea how to proceed. I know that the elements of $Q(\pi^4)$ are of the form $\frac{f(\pi^4)}{g(\pi^4)}$ where $f,g \in Q[x], g \neq 0.$

Moreover, is there a general way of finding the splitting fields of polynomials with transcendental elements? Please let me know.

Thanks!

Answer 1

You can factorise over $\mathbb{R} $ as $(x-\pi ^2) (x+\pi ^2). $ So you only need to adjoin $\pi ^2 $ so the splitting field is $ \mathbb{Q} (\pi ^4, \pi ^2)= \mathbb{Q}(\pi ^2)$.