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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- Feb 4th 2009, 03:47 AMpeteryellowthe 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. - Feb 4th 2009, 04:47 PMNonCommAlg
it's just rephrasing a definition: $\displaystyle a \in \mathbb{C}$ is transcendental if and only if there is no $\displaystyle 0 \neq p(x) \in \mathbb{Z}[x]$ such that $\displaystyle p(a)=0.$ now $\displaystyle W_a$ is not linearly independent if and only if $\displaystyle \sum_{j=0}^m n_ja^j=0,$ for some

$\displaystyle n_j \in \mathbb{Z}$ such that not all $\displaystyle n_j$ are equal to 0. hence if we let $\displaystyle p(x)=\sum_{j=0}^mn_jx^j \in \mathbb{Z}[x],$ we'll have $\displaystyle p(x) \neq 0$ and $\displaystyle p(a)=0.$