1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Why is Q(sqrt(2)) not isomorphic with Q(sqrt(3))

  1. Mar 2, 2008 #1
    1. The problem statement, all variables and given/known data

    why is Q[[tex]\sqrt{2}[/tex]] is not isomorphic to Q[[tex]\sqrt{3}[/tex]]?

    2. Relevant equations

    3. The attempt at a solution

    I do not know where to start?
  2. jcsd
  3. Mar 2, 2008 #2
    Assume you have an isomorphism [tex]f[/tex] from [tex]\mathbb{Q}[\sqrt{2}][/tex] to [tex]\mathbb{Q}[\sqrt{3}][/tex]. Think about what [tex]\sqrt{2}[/tex] can be mapped to. Hint: you can deduce that [tex]f(\sqrt{2})^2 = 2[/tex]
  4. Mar 2, 2008 #3
    Thanks for the help.
    If I assume there is an isomorphism from [tex]\mathbb{Q}[\sqrt{2}][/tex] to [tex]\mathbb{Q}[\sqrt{3}][/tex] then since [tex]\sqrt{2}[/tex] is not rational, it can only be mapped to [tex]\sqrt{3}[/tex]. This says that (a+b[tex]\sqrt{2}[/tex]) is mapped to (a+b[tex]\sqrt{3}[/tex]) but the map here is not a homomorphism since it fails this requirement: f(ab) = f(a)f(b).
    I am just curious if there is some characteristic that [tex]\mathbb{Q}[\sqrt{2}][/tex] but [tex]\mathbb{Q}[\sqrt{3}][/tex] does not? ( I guess not?)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Why is Q(sqrt(2)) not isomorphic with Q(sqrt(3))