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Field extension that is not normal

  1. May 1, 2013 #1
    I want to come up with an example of a field extension that is not normal, and seems to be difficult. All extension constructed in some obvious way tend to turn out normal.
     
  2. jcsd
  3. May 1, 2013 #2
    I don't know how you defined normal exactly (there are various equivalent definitions). But a very useful characterization is the following: if ##K## is normal over ##L## and if ##P(X)## is a irreducible polynomial over ##L## that has a root in ##K##, then ##P(X)## has all roots in ##K##.

    So take ##L=\mathbb{Q}##. Take some irreducible polynomial over ##\mathbb{Q}##. Adjoin a root of this polynomial to ##\mathbb{Q}## and see whether all roots are included.

    For examply, we know that ##X^3 - 2## is irreducible over ##\mathbb{Q}##. Do all roots of ##X^3-2## lie in ##\mathbb{Q}[\sqrt[3]{2}]##?
     
  4. May 1, 2013 #3
    oh dear...
     
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