Degree of field extension is infinite

Question

If we have the field extension $\mathbb{Q}\leq \mathbb{R}$, could you explain me why it stands that $[\mathbb{R}:\mathbb{Q}]=+\infty$ ??

Answer 1

One way to see this is that any finite extension of $\mathbb{Q}$ will be countable (being a finite dimensional vector space over $\mathbb{Q}$.

Since $\mathbb{R}$ is not countable, it cannot be a finite extension of $\mathbb{Q}$.

Answer 2

Another way: since $\pi$ is trascendental, no polynomial of $\Bbb Q[X]$ has $\Bbb R$ as its descomposition field.