ca_set_qqbar

A rational x gives an element of $\mathbb{Q}$, and an irrational x an element of a field $\mathbb{Q}(a)$ with one algebraic generator (never of a field with several generators: $\sqrt{2} + \sqrt{3}$ is not recognised as an element of $\mathbb{Q}(\sqrt{2}, \sqrt{3})$). A quadratic x is written as an element of $\mathbb{Q}(\sqrt{N})$, where $N$ is the discriminant of its minimal polynomial with small square factors removed by smooth factorisation, so that for example $\sqrt{12}$ and $1 + \sqrt{3}$ share the field $\mathbb{Q}(\sqrt{3})$. Higher-degree numbers are made generators of their own field $\mathbb{Q}(x)$.