How can i prove that is a ring isomorphism to
(They happen to be isomorphic as vector spaces, maybe that is what the poster was really asking).
i tried but i am not sure it is true i wish somebody will check the solution.
Let be a supposed ring isomorphism. Since and is onto, we must have This implies that Let , so 2= . Since the is irrational we must have either a=0 or b=0. Then we use either or irrational to conclude it is impossible to solve this equation with rational numbers.
Thus the suggestion is not well defined!!
canuhelp, did you read post #3?
Also, I dont think its a good idea to post the same question in multiple places. It is against forum rules.
What does the question ask you? "Give an isomorphism..." or "Is there an isomorphism..."?
I suspect it is the latter; the solution boils down to the question of "is a rational number", which i don't know off the top of my head.
Hint: Assume there exists an isomorphism. Some element maps to - what must it look like?
EDIT: I am entirely confused - I thought I was only the second person to post in this thread, after vemrygh. Thus my repetition of other peoples posts. I'm sure I read the entire thread first though!