This question already has an answer here:

Is $ mathbb {Q} ( sqrt {2}) cong mathbb {Q} ( sqrt {3}) $?
6 replies
This is probably a stupid question, but I can't solve it myself.
I think they are not, but I cannot formally prove it. One reason they are probably not isomorphic is that $ x ^ 22x1 in mathbb {Q} (x) subset mathbb {Q} ( sqrt2) (x) $ It has no roots in $ mathbb {Q} $, but it has its roots in $ mathbb {Q} ( sqrt2) $.
I am not sure if my argument is valid or not. Any suggestions / suggestions would be appreciated.