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Does this field contain i?

  1. Feb 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Is [tex]i \in \mathbb{Q}(\alpha)[/tex], where [tex]\alpha^3 + \alpha + 1 = 0[/tex]?

    2. Relevant equations

    3. The attempt at a solution

    Suppose [tex]i \in \mathbb{Q}(\alpha)[/tex]. Then the field [tex]\mathbb{Q}(i)[/tex] generated by the elements of [tex]\mathbb{Q}[/tex] and [tex]i[/tex] is an intermediate field, i.e.

    [tex]\mathbb{Q} \subset \mathbb{Q}(i) \subset \mathbb{Q}(\alpha)[/tex].

    But the degree [tex][\mathbb{Q}(i):\mathbb{Q}] = 2[/tex] does not divide the degree [tex][\mathbb{Q}(\alpha):\mathbb{Q}] = 3[/tex], so [tex]i \notin \mathbb{Q}(\alpha)[/tex].

    Is that right?
     
    Last edited: Feb 12, 2008
  2. jcsd
  3. Feb 12, 2008 #2

    Hurkyl

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    Sounds good. (You did verify that your polynomial is irreducible, right?)
     
  4. Feb 12, 2008 #3
    Great. Ahh, yes, that would certainly need to be shown. Thanks.

    I just wanted to make sure I'm getting these basic ideas down correctly, and not missing something completely obvious. We're just beginning Galois theory, and I'm using a couple supplementary texts because the one we use in class (Algebra, Michael Artin) is a bit tough for a first exposure to this stuff. It's great, though, after you've got a good handle on things. Thanks again!
     
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