I was trying to solve an olympiad type problem involving a nested radical of the form

[tex] \sqrt{a+b\sqrt{r}}.[/tex]

I had managed to discover that [tex] \sqrt{a^2- b^2r} [/tex] is an integer but it turned out the trick is to rewrite [tex] \sqrt{a+b\sqrt{r}} = c + d\sqrt{r}.[/tex]

Of course, one naturally asks if this is a specific incident or is there a general theory. This lead to digging up an article that I painstakingly photocopied from the library from back when photocopying was the norm. The article in question is by Susan Landau from 1994 in the Math. Intelligencer titled “How to tangle with a nested radical.”

A simplified version of Theorem 1 is this:

Let [tex] k [/tex] be a field extension of the rational numbers and [tex] a, b, r \in k [/tex] but [tex] \sqrt{r} \notin k [/tex]. Then

[tex] \sqrt{a^2- b^2r} \in k [/tex] is equivalent to [tex] \sqrt{a+b\sqrt{r}} \in k(\sqrt{s}, \sqrt{r}) [/tex] for some [tex] 0\ne s \in k [/tex].

For example, one may check that

[tex] \sqrt{5+2\sqrt{6}} =\sqrt{2}+\sqrt{3} [/tex].