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Math Help - Quadratic field extension of the rational number

  1. #1
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    Quadratic field extension of the rational number

    about the set of number W={a+(2^(1/2))b: a,b∈Q}
    Q is set of rational number.

    a)how to show x=a+(2^(1/2))b∈Q if and only if b=0?
    b)how to show W is countable?
    c)how to show that between any two distinct real number is an element of W?
    d)how to show W is an ordered field?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Two little hints:

    a) Prove that if a+b\sqrt{2}\in \mathbb{Q} and b\neq 0 then, \sqrt{2} would be rational (contradiction).

    b) \mathbb{Q} is countable and \#(W)=\#(\mathbb{Q}\times \mathbb{Q}) .

    Try the rest.
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