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Math Help - Field extension question

  1. #1
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    Field extension question

    I'm trying to answer this question:

    "Let M:L:K be finite field extensions. When M is not normal over K, give four examples to show that this gives us no information about the normality of M over L or of L over K. What are the possibilities if M is normal over K?"

    Does anyone have any ideas?
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  2. #2
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    Hi. Can you verify if I'm right?

    \mathbb{Q}\subset\mathbb{Q}(\sqrt{2})\subset\mathb  b{Q}(\sqrt[4]{2}) (normal/normal)
    \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{Q}(\sqrt[6]{2}) (not normal/normal)
    \mathbb{Q}\subset\mathbb{Q}(\sqrt{2})\subset\mathb  b{Q}(\sqrt[6]{2}) (normal/not normal)
    \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{Q}(\sqrt[9]{2}) (not normal/not normal)

    Now assume M is a normal extention over K, and K\subseteq L\subseteq M
    M is normal over K iff given an algebraic closure C of M, any K-isomorphism from M to another subfield of C is a K-automorphism of M. Since a L-isomorphism from M to another subfield of C is also a K-isomorphism, then it is a L-automorphism of M and M is a normal extension over L.

    Finally, consider:

    \mathbb{Q}\subset\mathbb{Q}(j)\subset\mathbb{Q}(\s  qrt[3]{2},j)
    \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{Q}(\sqrt[3]{2},j)

    \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2},j) is a normal extension, but what about \mathbb{Q}\subset\mathbb{Q}(\sqrt[3]{2}) and \mathbb{Q}\subset\mathbb{Q}(j)?
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  3. #3
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    Quote Originally Posted by KSM08 View Post
    I'm trying to answer this question:

    "Let M:L:K be finite field extensions. When M is not normal over K, give four examples to show that this gives us no information about the normality of M over L or of L over K. What are the possibilities if M is normal over K?"

    Does anyone have any ideas?

    K=\mathbb{Q}\leq L=\mathbb{Q}(\sqrt{2})\leq M=\mathbb{Q}(2^{1\slash 4}) . M\slash K isn't normal, but M\slash L\,,\,L\slash K are.

    K=\mathbb{Q}\leq L=\mathbb{Q}(2^{1\slash 4})\leq M=\mathbb{Q}(2^{1\slash 8}) . M\slash K\,,\,L/K aren't normal, but M\slash L is.

    Now you try other cases.

    Tonio
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