Is there an explicit basis for the algebraic numbers as a vector space over the rationals?



Basis for the Algebraic numbers over the rationals
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.


