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Math Help - algebraic numbers

  1. #1
    Senior Member I-Think's Avatar
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    algebraic numbers

    Let F\subset{E}. both E and F are fields, \alpha, \beta\in{E}. If \alpha+\beta and \alpha\beta are algebraic over F, show that \alpha and \beta are algebraic
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  2. #2
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    Re: algebraic numbers

    consider the field K = F(α+β,αβ) which is a finite extension of F since α+β and αβ are algebraic over F.

    note that α,β are the roots of x2 - (α+β)x + αβ in K[x].

    thus [F(α,β):F] = [F(α,β):K][K:F], since both factors are finite, so is their product.

    thus the subfields F(α) and F(β) of F(α,β) must have finite degree so α,β are algebraic.
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  3. #3
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    Re: algebraic numbers

    Nice proof, regardless!
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