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Examples of normal and non-normal extensions of Q?

  1. Nov 27, 2011 #1
    Hi, I'm really struggling to find examples (with proofs) of the following:

    1) For each n>2 give an example of a non-normal extension of Q of degree n.

    2) Give examples of normal extensions of Q of degrees 3,4 and 5.

    3) Show that for any positive integer n, there exists a normal extension of Q of degree n.

    Any help would be much appreciated!
  2. jcsd
  3. Nov 27, 2011 #2
    So an extension [itex]\mathbb{Q}\subseteq K[/itex] is normal if for all [itex]\alpha\in K[/itex] the minimal polynomial of [itex]\alpha[/itex] splits in K. Or equivalently if K is the splitting field of a polynomial in [itex]\mathbb{Q}[/itex].

    So, can you adjoin a number [itex]\alpha[/itex] to [itex]\mathbb{Q}[/itex] such that the minimal polynomial doesn't split?? This answers (a).
  4. Nov 27, 2011 #3


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    for 2) remember that the algebraic closure of Q is not a subfield of R, so you need to look for some complex numbers that make this happen. i suggest looking on the unit circle, perhaps?
  5. Nov 27, 2011 #4
    But adjoining one number will surely not give you an extension of degree n as required in the question?
  6. Nov 27, 2011 #5
    It might, for example, adjoining [itex]\sqrt[3]{2}[/itex] gives you a nonnormal extension of degree 3.
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