TITLE: Weird subfields of $\Bbb{R}$
QUESTION [16 upvotes]: I found this problem, and I can't get an answer to it:

Prove that there are subfields of $\Bbb{R}$ that are
a) non-measurable.
b) of measure zero and continuum cardinality.

I can't seem to imagine how to construct such subfields of $\Bbb{R}$.

REPLY [2 votes]: You can try the answers (including mine) for a question at MathOverflow... https://mathoverflow.net/questions/27352/a-question-about-fields-of-real-numbers/27358#27358