Isomorphism: is field of reals isomorphic to field of rtnls?

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jessica098

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I need help answering the following question:

Is the field of real numbers isomorphic to the field of rational numbers?

My teacher gave me a hint, and that is to assume that there IS an isomorphism from the reals to the rationals and work with an equation such as the sqrt2xsqrt2 = 2.

Any help would be GREATLY appreciated! THANKS!
 
If there existed an isomorphism F between R and Q then F is, among other things, bijective. You can't have a bijective function between two sets with different cardinality.
 
The cardinality argument works - if you've learned about cardinality.

If you want to follow your teacher's hint, then let x be the image in Q of the sqrt(2). It's easy to prove that x^2=2, and your textbook or notes surely has proofs that this x can't be rational?
 
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