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## Main Question or Discussion Point

Hello all,

I am studying Algebra and in the chapter where Galois theory is introduced, I

see the following exercise:

"Prove that Q(sqrt(2)) and Q(sqrt(3)) are not isomorphic"

Well, It seems that I am a bit behind because I really don't get it... :(

I mean, I'm sure that this is the case, since it is a question in the book

(and surely 'not' is not a typo!!), but these are vector spaces over Q,

both of dimension 2, so shouldn't they be isomorphic by sending

sqrt(2) to sqrt(3) and any rational number to itself?!

What do I miss here?

Thanks a lot in advance..

I am studying Algebra and in the chapter where Galois theory is introduced, I

see the following exercise:

"Prove that Q(sqrt(2)) and Q(sqrt(3)) are not isomorphic"

Well, It seems that I am a bit behind because I really don't get it... :(

I mean, I'm sure that this is the case, since it is a question in the book

(and surely 'not' is not a typo!!), but these are vector spaces over Q,

both of dimension 2, so shouldn't they be isomorphic by sending

sqrt(2) to sqrt(3) and any rational number to itself?!

What do I miss here?

Thanks a lot in advance..

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