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Math Help - the complex numbers C as a module over integers Z

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    the complex numbers C as a module over integers Z

    Consider the complex numbers C as a module over integers Z, and let a be a complex number.
    Consider the subset W_a = {1,a,a^2,...,a^n,...}
    of the Z-module C. Then I want to show that
    W_a is linearly independent iff a is a transcedent number.
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    Quote Originally Posted by peteryellow View Post

    Consider the complex numbers C as a module over integers Z, and let a be a complex number.
    Consider the subset W_a = {1,a,a^2,...,a^n,...}
    of the Z-module C. Then I want to show that
    W_a is linearly independent iff a is a transcedent number.
    it's just rephrasing a definition: a \in \mathbb{C} is transcendental if and only if there is no 0 \neq p(x) \in \mathbb{Z}[x] such that p(a)=0. now W_a is not linearly independent if and only if \sum_{j=0}^m n_ja^j=0, for some
    n_j \in \mathbb{Z} such that not all n_j are equal to 0. hence if we let p(x)=\sum_{j=0}^mn_jx^j \in \mathbb{Z}[x], we'll have p(x) \neq 0 and p(a)=0.
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