Let $f(x)$ be an algebraic function over the field $\mathcal F$ of algebraic numbers over $\mathbb{Q}$. Suppose that $r \in \mathcal F$. Does the discriminant of $f(r)$ divide the discriminant of $f(f(r))$?
Example: take $r$ to be any algebraic number and $$f(x) = (x + \sqrt{4 + x^2} )/2 = [x,x,x, \ldots]$$ Writing $D$ for discriminant, in this example and many others, something much stronger may be true: $(D(f(r)))^2 | D(f(f(r))).$ Are there known conditions (on $f$ and $r$) for this stronger divisibility?
Thanks, GNiklasch. Yes, I mean "discriminant" to be the field discriminant, as given by Mathematica's "NumberFieldDiscriminant". Let's assume that $\mathbb{Q}(f(f( r)))$ contains $\mathbb{Q}(f( r))$ as a subfield -- maybe later consider other possibilities.