TITLE: Basis for the Algebraic numbers over the rationals
QUESTION [9 upvotes]: Is there an explicit basis for the algebraic numbers as a vector space over the rationals?

REPLY [12 votes]: Every computable field which is an algebraic extension of the rationals $\mathbb{Q}$ has a computable basis (as a vector space over $\mathbb{Q}$).  The idea is to build up this basis by recursion:  let $F_0 = \mathbb{Q}$, with basis $B_0=$ {$1$}, and, given a basis for $F_s$ over $\mathbb{Q}$, find the least element $x$ of $F$ whose minimal polynomial over $F_s$ has degree $d > 1$.  This can be done because $F_s$ has a splitting algorithm, enabling one to recognize irreducible polynomials in $F_s[X]$, uniformly in $s$.  Then $C_s=${$ 1, x, x^2, x^3,...x^{d-1}$} is a basis for $F_{s+1}=F_s[x]$ over $F_s$, and a basis $B_{s+1}$ for $F_{s+1}$ over $\mathbb{Q}$ is given by all products of one element of $C_s$ with one element of the basis $B_s$ for $F_s$ over $\mathbb{Q}$.  Since $1$ lies in $C_s$, we have $B_s \subset B_{s+1}$, and the union (over $s$) of all these $B_s$ will be a basis $B$ for $F$ over $\mathbb{Q}$.  The same construction would work with any finite field in place of $\mathbb{Q}$, and indeed over every c.e. ground field which has a splitting algorithm.
However, when $F$ is transcendental over $\mathbb{Q}$ (especially when it has infinite transcendence degree), this can no longer be done.  In this case a computable vector-space basis follows if there exists a computable transcendence basis for $F$ over $\mathbb{Q}$, but not all computable field extensions of $\mathbb{Q}$ have computable transcendence bases.  Metakides and Nerode produced a computable field of infinite transcendence degree over $\mathbb{Q}$ which has no computable vector-space basis over $\mathbb{Q}$.  One could probably also build a computable field $\mathbb{Q}$ with computable vector-space basis over $\mathbb{Q}$, but with no computable transcendence basis over $\mathbb{Q}$.
A discussion of splitting algorithms, over $\mathbb{Q}$ and over finitely generated extensions, appears in Harold Edwards' book Galois Theory.  Fried & Jarden is another good source.