Exercise 13.2 from Geometry: Euclid and Beyond by Hartshorne: Show that any quantity obtainable from the rational numbers by a finite number of operations $+, -, \cdot, \div, \sqrt{ }$ can be written in a standard form $r \cdot A$, where $r \in \mathbb{Q}$ is a arational number and $A$ is an expression involving only integers, $+, -, \cdot$, and $\sqrt{ }$.
Here are my thoughts on this:
$+, -, \cdot, \sqrt{ }$ preserve the standard form, so we only need to show that $\div$ also preserves it. A simpler and possibly sufficient problem to solve is, to show that $1 / \sum_i^n A_i \sqrt{B_i}$ can be rationalized for $A_i, B_i \in \mathbb{Z} \setminus \{0\}$, and $\{B_i\}_i$ distinct, by multiplying something of the structure $\sum_i^n C_i \sqrt{B_i} + \sum_{i<j} C_{ij} \sqrt{B_i B_j} + \dots + C_{1,2,\dots,n} \sqrt{B_1 B_2 \dots B_n}$. This becomes a complicated set of linear equations to solve. The case of $n=2$ is certainly easy. The case of $n=3$ and $B_i = 1$ is already very messy, and this is not even the general inductive step. Any suggestions?