I want to construct the real algebraic numbers from $\mathbb{Q}$ in a manner that sort of "looks like" the construction of the complex numbers from the reals, in a superficial manner. I can't define a single "irrational unit", but I instead tried to start with the set $\{ \sqrt{z} \mid z \in \mathbb{Q} \}$ and build the algebraic numbers from there.
This is what I'm thinking:
Let $A_{0} = \{ q + p \sqrt{z} \mid q, p, z \in \mathbb{Q} \}$.
In general, let $A_{n} = \{ q + p \sqrt{z} \mid q, p, z \in \mathbb{Q} \cup A_{0} \cup \dots \cup A_{n-1} \}$.
Let $A = \cup_{i=0}^{\infty} A_{i}$.
Is $A = \mathbb{A}$? If not, what's missing, and if so, can this description be simplified in some way?
Thanks all.
These are called the constructible numbers, so named because they are the numbers you can construct via Euclidean geometry with a compass and straightedge, starting from a unit interval.
Adjoining all square roots isn't enough to produce the algebraic numbers; you can never produce $\sqrt[3]{2}$ in this fashion.
Adjoining $n$-th roots isn't enough either; for example, there are explicitly known polynomials of degree five whose roots cannot be expressed in that fashion. (i.e. the "insolvability" of the quintic)