Thread: 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.

Originally Posted by peteryellow

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: is transcendental if and only if there is no such that now is not linearly independent if and only if for some
such that not all are equal to 0. hence if we let we'll have and