Results 1 to 6 of 6

Math Help - Showing two fields are not isomorphic

  1. #1
    Junior Member
    Joined
    May 2011
    Posts
    59

    Showing two fields are not isomorphic

    I am trying to show that \mathbb{Q}(\sqrt{2}) and \mathbb{Q}(\sqrt{3}) are not isomorphic.

    My question is this... Isomorphisms preserve subfields (including, in this case, \mathbb{Q}), and I WANT to say something like ' \mathbb{Q} must be preserved so the mapping acts as the identity on \mathbb{Q} but then it must act as the identity on the whole field, but then clearly it is not an isomorphism so \mathbb{Q}(\sqrt{2}) and \mathbb{Q}(\sqrt{3}) cannot be isomorphic.'

    But, is any of this true, or even a useful way of thinking about it? Sadly, I suspect not...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,696
    Thanks
    1467

    Re: Showing two fields are not isomorphic

    why is f(a+ b\sqrt{2})= a+ b\sqrt{3} not an isomorphism?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Showing two fields are not isomorphic

    Quote Originally Posted by AlexP View Post
    I am trying to show that \mathbb{Q}(\sqrt{2}) and \mathbb{Q}(\sqrt{3}) are not isomorphic.

    My question is this... Isomorphisms preserve subfields (including, in this case, \mathbb{Q}), and I WANT to say something like ' \mathbb{Q} must be preserved so the mapping acts as the identity on \mathbb{Q} but then it must act as the identity on the whole field, but then clearly it is not an isomorphism so \mathbb{Q}(\sqrt{2}) and \mathbb{Q}(\sqrt{3}) cannot be isomorphic.'

    But, is any of this true, or even a useful way of thinking about it? Sadly, I suspect not...
    Suppose that f:\mathbb{Q}(\sqrt{2})\to\mathbb{Q}(\sqrt{3}) is an isomorphism then f(\sqrt{2})^2-2=f\left(\sqrt{2}^2-2\right)=0 and so there exists some y\in\mathbb{Q}(\sqrt{3}) such that y^2-2=0.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    May 2011
    Posts
    59

    Re: Showing two fields are not isomorphic

    HallsofIvy: it is not a homomorphism. When you multiply the square roots cancel each other so you end up with a 2 on one side and a 3 on the other.

    Drexel28, I understand the point of showing that there is no number that squares to 2 in \mathbb{Q}(\sqrt{3}), but can you explain the f(\sqrt{2})^2-2 = f(\sqrt{2}^2-2) part?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Showing two fields are not isomorphic

    Quote Originally Posted by AlexP View Post
    HallsofIvy: it is not a homomorphism. When you multiply the square roots cancel each other so you end up with a 2 on one side and a 3 on the other.

    Drexel28, I understand the point of showing that there is no number that squares to 2 in \mathbb{Q}(\sqrt{3}), but can you explain the f(\sqrt{2})^2-2 = f(\sqrt{2}^2-2) part?
    Since we are dealing with fields every morphism is unital, i.e. if f:\mathbb{Q}(2)\to\mathbb{Q}(3) is an isomorphism then f(1)=1 and so f(2)=f(1+1)=f(1)+f(1)=2 thus f\left(\sqrt{2}^2-2\right)=f\left(\sqrt{2}^2\right)-f(2)=f\left(\sqrt{2}\right)^2-2.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    May 2011
    Posts
    59

    Re: Showing two fields are not isomorphic

    Ah, right. I didn't see how f(2)=2 but that explains it. Thanks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. showing D12 has a subgroup isomorphic to d4
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 4th 2011, 11:39 AM
  2. [SOLVED] Prove that both rings are fields but that they are not isomorphic.
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: March 30th 2011, 07:28 AM
  3. vector fields - graphing gradient fields
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 20th 2010, 05:53 PM
  4. showing that groups are isomorphic
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 30th 2009, 10:36 PM
  5. Isomorphic Fields
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 13th 2006, 09:50 PM

Search Tags


/mathhelpforum @mathhelpforum