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Math Help - transcendence base

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    transcendence base

    Let be the quotien field of
    Exhibit a transcendence base for over , and express explicitly as an algebraic extension over .
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  2. #2
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    Quote Originally Posted by Stiger View Post
    Let be the quotien field of
    Exhibit a transcendence base for over , and express explicitly as an algebraic extension over .
    let f=x_1^2 + x_2^2 + x_3^2 - 1 and v_j=x_j + <f>, \ \ 1 \leq j \leq 5. the claim is that S=\{v_2,v_3,v_4,v_5 \} is a transcendence basis for F=\mathbb{C}(S,v_1) over \mathbb{C}: since v_1^2+v_2^2 +v_3^2-1=0, the element v_1 is

    algebraic over \mathbb{C}(S) and thus F is algebraic over \mathbb{C}(S). so to prove that S is a trancendence basis for F over \mathbb{C}, we only need to show that S is algebraically independent over \mathbb{C}. so suppose that

    the elements of S are algebraically dependent over \mathbb{C}. then there exists 0 \neq g \in \mathbb{C}[y_1, y_2, y_3, y_4] such that g(v_2,v_3,v_4,v_5)=0, i.e. g(x_2,x_3,x_4,x_5) \in <f>, which is obviously impossible because

    every non-zero element of <f> involves x_1 but g does not. \Box
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