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!

Is the field extension normal?

  1. Oct 30, 2009 #1
    I've been asked to find out if some field extensisons are normal. I want to know if I'm thinking about these in the right way.

    For Q(a):Q

    I first find the minimal polynomial for a in Q[a]. Then I look at all zeros of that polynomial. If all of the zeros are in Q(a) the extension is normal.


    Example:

    Q(1+i):Q

    1+i = x
    -1 = x^2-2x+1

    x^2-2x+2 is irreducible over Q and the minimal polynomial of 1+i.

    the zereos are: 1+i, 1-i

    they are both in Q(1+i) so this is a normal extension.


    Correct?
     
  2. jcsd
  3. Oct 30, 2009 #2
  4. Oct 30, 2009 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, it's normal. Do you know why field extensions might not be normal?
     
  5. Oct 30, 2009 #4
    Q(2^(1/3)):Q is not normal since the minimal polynomial has two imaginary roots that are not in Q(2^(1/3)). Is that the right idea?
     
  6. Oct 30, 2009 #5
    I forgot to say that 2^(1/3) is the real cube root of two.
     
  7. Oct 30, 2009 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, I think so. In your first example, including one root automatically includes the other. In the second it doesn't include the other roots.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Is the field extension normal?
  1. Extension fields (Replies: 2)

  2. Field Extensions (Replies: 26)

  3. Extension Field (Replies: 2)

Loading...