Is the field extension of $\mathbb{R}$ over $\mathbb{Q}$ a countable infinity or uncountable infinity?

Question

In looking at other questions/answers here on MSE, $[\mathbb{R}:\mathbb{Q}]$ is infinite, but they didn't specify whether it was a countable infinity or uncountable infinity. I would guess uncountable since even a union of countably infinite many copies of a countable set like $\mathbb{Q}$ is still countably infinite, and the irrationals $\mathbb{R} - \mathbb{Q}$ are uncountable, but I'm not sure if a set-theoretic argument like that would transfer over to a statement about field extensions. Thanks for any advice.