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Need help understanding splitting fields

  1. Feb 21, 2015 #1
    My textbook is going through an example on splitting fields. It asked to find a splitting field for x^4 - 6x^2 - 7 over the rational numbers. This polynomial factors to (x^2 - 7)*(x^2+1) which has roots of 7^(1/2) and i. So i figured the extension field E we are looking for is Q(i)(7^(1/2)), but my textbook jumps straight to Q(i)(2^(1/2).

    is the squareroot of 7 an element of the simple extension of the rational numbers with the square root of two? I can't imagine it being (yet i can't imagine many things that are..).

    Any mathamaverick wanna shed some light on my situation?
     
  2. jcsd
  3. Feb 21, 2015 #2

    WWGD

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    Doesn't it follow, if ##2^{1/2}## is in a field extension of the rationals that ##(7/2)( 2^{1/2}) ## is also on the field, e.g., by closure under multiplication? Your result is correct if you re not looking for a minimal extension.
     
  4. Feb 21, 2015 #3
    Okay, what am i missing? How does (7/2)*(2^(1/2)) show that the finite extensions of 7^(1/2) reduces to 2^(1/2)?
     
  5. Feb 22, 2015 #4

    WWGD

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    Ah, sorry, I misread, let me think again.
     
  6. Feb 24, 2015 #5

    HallsofIvy

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    It looks to me like a misprint. No, the extension fields containing p^{1/2} and q^{1/2}, for p and q prime, are NOT the same.
     
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