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Math Help - Field extension problem

  1. #1
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    Field extension problem

    Show that  \sqrt {3} is not in  \mathbb {Q} [ ^4 \sqrt {2} ]

    Proof so far:

    Let x = ^4 \sqrt {2}

    Assume to the opposite that  \sqrt {3} = a + bx+cx^2 +dx^3 \ , \ a,b,c,d \in \mathbb {Q}

    Now, I want to take the trace of both side, but I'm really having problem with field trace, I don't know how to take the trace of neither side, are there any references I can find online that reteach me what trace is and how to compute it?

    Thank you very much!
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  2. #2
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    You can prove this by showing, \sqrt{3} = a+b\sqrt[4]{2}+c\sqrt[4]{4}+d\sqrt[4]{8} is impossible by squaring sides.
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  3. #3
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    thanks, but it is just that I want to learn more about field trace, I'm really really lost in this thing.
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