Can the interval $[0,1]$ be made into a field?

Question

After some cups of coffee with a friend we come up with a non-trivial question to our knowledge and it reads as follows :

Is is possible to define the operations of sum and product on $[0,1]$ so that makes it a field ?

As mentioned, this question is most likely beyond of my reach. So, could you give me some lead to solve this problem ?

Answer 1

Is there a field with the same cardinal as $[0,1]$? Yes: take $(\Bbb R,+,.)$, for instance. So, take a bijection $b\colon[0,1]\longrightarrow\Bbb R$ and define on $[0,1]$:

• the additive identity is $b^{-1}(0)$;
• the multiplicative identity is $b^{-1}(1)$;
• $x+y=b^{-1}\bigl(b(x)+b(y)\bigr)$;
• $x.y=b^{-1}\bigl(b(x).b(y)\bigr)$.

And now you have a field $([0,1],+,.)$, which is isomorphic to $(\Bbb R,+,.)$.