Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Isomorphic Subfields

  1. Jan 21, 2014 #1
    I've got this weird thought in my head that isomorphic subfields of any field are equal. I'm having trouble believing this and I'm trying to come up with a counterexample to quell my head. But, I'm having a lot of trouble coming up with such a counterexample, and I don't know why.

    Can anyone think of a quick counterexample?

    So far, I've thought of the field isomorphic to ##\mathbb{Z}/(2)## generated by ##0,1\in \mathbb{F}_4##. But, when I thought about it more, no other subfield is isomorphic to ##\mathbb{Z}/(2)## in ##\mathbb{F}_4##, since ##0## and ##1## would necessarily be in the subfield. Then I thought about how ##\mathbb{R}\cong \mathbb{R}i## as vector spaces, but clearly not as rings (##\mathbb{R}i\not\ni 1##).

    As usual, help is greatly appreciated.
     
  2. jcsd
  3. Jan 21, 2014 #2

    jgens

    User Avatar
    Gold Member

    Notice that the inclusion k(x2,x3,...) ⊂ k(x1,x2,x3,...) is proper, yet these fields are isomorphic.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Isomorphic Subfields
  1. Fields and Subfields (Replies: 11)

  2. Subfields of a field (Replies: 2)

  3. Subrings and Subfields (Replies: 14)

  4. Are they isomorphic? (Replies: 4)

Loading...