Compute $[\mathbb{Q}(\pi):\mathbb{Q}(\pi^2)]$ and $\{\mathbb{Q}(\pi):\mathbb{Q}(\pi^2)\}$

Question

The problem asked to compute $[\mathbb{Q}(\pi):\mathbb{Q}(\pi^2)]$ and $\{\mathbb{Q}(\pi):\mathbb{Q}(\pi^2)\}$, where the first one is the degree of the extension and the second one is the index of the splitting field. I don't think the first one even exists since $\mathbb{Q}(\pi)$ is not a finite extension itself. I don't know how to handle the second one.

Any help would be appreciated!

Answer 1

Hint:

The polynomial $\;x^2-\pi^2\in\Bbb Q(\pi^2)[x]\;$ vanishes at $\;x=\pi\;$ , so $\;[\Bbb Q(\pi):\Bbb Q(\pi^2)]\le2\;$ ... assuming you meant the extension $\;\Bbb Q(\pi)/\Bbb Q(\pi^2)\;$