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Math Help - Finite Extensions

  1. #1
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    Finite Extensions

    I was thinking about this and was not able to prove this. I believe that if this is true then it might be useful in field theory.

    If, F\leq E are fields, and \alpha,\beta\in E algebraic over F and [F(\alpha):F]=[F(\beta):F].
    Prove (or disprove):
    F(\alpha)=F(\beta).
    ???
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  2. #2
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    Quote Originally Posted by ThePerfectHacker
    I was thinking about this and was not able to prove this. I believe that if this is true then it might be useful in field theory.

    If, F\leq E are fields, and \alpha,\beta\in E algebraic over F and [F(\alpha):F]=[F(\beta):F].
    Prove (or disprove):
    F(\alpha)=F(\beta).
    ???
    I might have the wrong idea here, because I'm not sure how you are using \leq, but my thought is:

    Let F= \mathbb{Q} and E= \mathbb{Q} [ \sqrt2 ,\sqrt3 ] . \sqrt2 and \sqrt3 are both algebraic over \mathbb{Q} and  [ \mathbb{Q} ( \sqrt2 ): \mathbb{Q} ] = [ \mathbb{Q} (\sqrt3 ): \mathbb{Q} ] .

    However,  \mathbb{Q} ( \sqrt2 ) \neq \mathbb{Q} ( \sqrt3 ) .

    -Dan
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    You said well, thank you.
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