My question is for formulation or notation of numerators and denominators of variable-containing algebraic irrational terms (radicals) in general.
I'm looking for formulations to talk about variable-containing algebraic irrational terms
- whose numerator and denominator don't contain the same factor (are coprime) and
- whose numerator doesn't contain a certain factor.
But can numerator and denominator of any given variable-containing algebraic irrational term be defined uniquely?
Is there a suitable normal form for variable-containing algebraic irrational terms in algebraic simplification?
Clearly, all terms with negative exponents must be transformed into terms with positive exponents.
A very very simple example:
$$-1+\sqrt{1+\sqrt{\frac{1+x}{1-x}}}$$ $$=-1+\sqrt{1+\frac{\sqrt{1+x}}{\sqrt{1-x}}}$$ $$=-1+\sqrt{\frac{\sqrt{1-x}+\sqrt{1+x}}{\sqrt{1-x}}}$$ $$=-1+\frac{\sqrt{\sqrt{1-x}+\sqrt{1+x}}}{(1-x)^\frac{1}{4}}$$ $$=\frac{-(1-x)^\frac{1}{4}+\sqrt{\sqrt{1-x}+\sqrt{1+x}}}{(1-x)^\frac{1}{4}}$$
I want to speak about equations of the form
$$A(x)=0,$$
wherein $A$ is an algebraic function (over $\mathbb{Q}$, $\mathbb{C}$ or over any other field). I want to formulate that I want to split the largest possible denominator by multiplication with it. But how can I express/formulate this if the terms "numerator" and "denominator" aren't defined uniquely for variable-containing algebraic irrational terms?
Addition:
A general algebraic irrational term has some analogies to a finite continued fraction. Can all finite continued fractions be transformed into an ordinary fraction? How can I formulate mathematically correct that the numerator and denominator of the resulting ordinary fraction should be coprime?