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**Mathstud28** Hello everyone! Once again there are no answers in the book so this is just another check just to make sure. Thank you very much in advance.

Question: "Are the set of all algaebraic numbers countable? Justify your respose"

I will attempt to prove that the algaebraic numbers are countable. Once again for my sake let $\displaystyle \mathbb{A}$ be the set of all algaebraic numbers.

Answer: Let $\displaystyle \zeta$ be an algaebraic number. Then by definition there exists a polynomial $\displaystyle p(x)=a_0+a_1x+\cdots+a_n(x)\quad{a}_0,a_1,\cdots,a _n\in\mathbb{Z}$, such that $\displaystyle p\left(\zeta\right)=0$. Therefore let us define the number $\displaystyle \zeta$ by the n-tuple $\displaystyle \left(a_0,a_1,\cdots,a_n\right)=t_1$. So now that we have shown that each algaebraic number may be expressed as a n-tuple we may state that $\displaystyle \mathbb{A}\equiv\left(t_1,t_2,\cdots,t_m\right)$ where $\displaystyle t_m\equiv\left(a_0,a_1,\cdots,a_n\right)$. So now since we have shown that the set of all algaebraic numbers may expressed as a set of n-tuples with each element of the n-tuple being an element of the integers, a countable set, we may conclude that the algaebraic numbers are countable $\displaystyle \quad\blacksquare$

I know its relatively simple, but I just want to sure I am doing this right.