Constructing an Algebraically Closed Space

Question

How would one construct the simplest -- or canonical, if a canonical construction exists -- countably infinite algebraic closure of the natural numbers?

Answer 1

One construction is simply to take all algebraic elements of $\mathbb{C}$ But a more general method is to enumerate all polynomials with integer coefficients, $p_n(x)$ and define $F_{n+1}$ to be $F_n(\alpha_1, \ldots ,\alpha_k)$ where $\alpha_1, \ldots ,\alpha_k$ are the roots of $p_{n+1}(x)$. Then $F=\bigcup F_n$ will be algebraically closed. For if $\beta$ is algebraic over $F$ there is some $n$ such that $\beta$ is algebraic over $F_n$ and so the extension $F_n(\beta)/F_0$ is finite dimensional and so $\beta$ is algebraic over $F_0$ and will have been included in the construction.