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.
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: $\displaystyle a \in \mathbb{C}$ 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 $\displaystyle p(a)=0.$