Two minor questions about a transcendental number over $\Bbb Q$

Question

I'm now painfully studying abstract algebra. I want to ask two minor questions to clarify my concepts, though they would be somehow silly. We know that $\pi$ is transcendental over $\Bbb Q$, hence $\Bbb Q[\pi]$ is an integral domain, and $\Bbb Q[\pi]\subsetneq\Bbb Q(\pi)$. Then I wonder if there is a field $F^\star$ such that $\Bbb Q(\pi)\subsetneq F^\star\subsetneq \Bbb R$? And is there any relationship between $\Bbb Q(\pi)$ and say, $\Bbb Q(e)$? ($e$ is the Euler constant)

Answer 1

1. $\mathbb{Q}(\pi)\varsubsetneq\mathbb{Q}\bigl(\sqrt\pi\bigr)\varsubsetneq\mathbb R$.
2. $\mathbb{Q}(\pi)\simeq\mathbb{Q}(e)$.