How can i prove that is a ring isomorphism to
I disagree with TheAbstractionist because is not isomorphic to as fields (so they cannot be isomorphic as fields). Now since are algebraic integers it means . Thus, asking for a ring isomorphism between and is asking for an isomorphism between and which does not exist.
(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.
By the suggestion above:
and
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!