Q(sqrt(2)) and Q(sqrt(3)) not isomorphic?

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Discussion Overview

The discussion revolves around the isomorphism between the fields Q(sqrt(2)) and Q(sqrt(3)), specifically addressing whether these two fields are isomorphic. The context includes concepts from Galois theory and field theory, focusing on the properties of these fields as vector spaces and fields.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant initially suggests that Q(sqrt(2)) and Q(sqrt(3)) should be isomorphic as vector spaces over Q, as both have dimension 2.
  • Another participant clarifies that while they are isomorphic as vector spaces, they are not isomorphic as fields because the proposed isomorphism does not respect the field product.
  • A later reply emphasizes the need to show that no field isomorphism can exist between Q(sqrt(2)) and Q(sqrt(3)), noting that any isomorphism must fix Q and that sqrt(2) cannot be sent to a rational number.
  • It is proposed that the reasoning can be generalized to any square-free integers m and n, asserting that Q(sqrt(m)) and Q(sqrt(n)) are also nonisomorphic.

Areas of Agreement / Disagreement

Participants generally agree that Q(sqrt(2)) and Q(sqrt(3)) are not isomorphic as fields, but there is discussion about the nature of isomorphisms and the conditions under which they hold.

Contextual Notes

The discussion highlights the distinction between vector space isomorphism and field isomorphism, with emphasis on the necessity for isomorphisms to respect field operations. There are unresolved aspects regarding the rigorous proof of nonisomorphism.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of algebra, particularly those studying Galois theory and field theory, as well as anyone interested in the properties of number fields.

geor
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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..
 
Last edited:
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Ooops! I think I see it now..
They are isomorphic as vector spaces but not as fields, right?
The isomorphism I said above does not respect the product..

That's it, right?!
 
Right. (Have you yet shown there isn't a field isomorphism?)
 
What more is true, is that given any square free intgers m and n, Q(sqrt(m)) and Q(sqrt(n)) are nonisomorphic. Intution serves right when you say that it "does not respect the product" but being more rigorous, show that no ismorphism can possibly exist between the two fields by first showing that any isomorphism fixes Q and that sqrt 2 (in this specific case) cannot be sent to any rational number, ie. sqrt 2 is sent to a+b*sqrt 3 for some nonzero rational b. This proof easily generalizes to square free m and n.
 
Thanks a lot!
 

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