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.
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1As noted here, the dimension must be $2^{\aleph_0}$. – Arturo Magidin Dec 27 '20 at 23:02
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2Also here are explicit examples of continuum-sized $\mathbb{Q}$-linearly independent subsets of $\mathbb{R}$. – Arturo Magidin Dec 27 '20 at 23:04
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That is perfect, thank you. – Hank Igoe Dec 28 '20 at 01:57