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Homework Help: 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

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    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

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    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.
     
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