Consider the set $\mathbb{Q}^\sqrt{}$ of real numbers that can be constructed by applying finitely many of the five operations $+$, $-$, $\cdot$, $/$ and $\sqrt{}$ to a rational number. Examples would be $\sqrt{7} - \sqrt{5} - \sqrt{3}$ or $\sqrt{3+\sqrt{2}}-3$.

I would like to find a procedure (an algorithm) which determines of a given number in $\mathbb{Q}^\sqrt{}$ if it is zero or not.

If it is impossible to give such a procedure, I would like to know why.

Consider the set $\mathbb{Q}^\sqrt{}$ of real numbers that can be constructed by applying finitely many of the five operations $+$, $-$, $\cdot$, $/$ and $\sqrt{}$ to a positive rational number. Examples would be $\sqrt{7} - \sqrt{5} - \sqrt{3}$ or $\sqrt{3+\sqrt{2}}-3$.

I would like to find a procedure (an algorithm) which determines of a given number in $\mathbb{Q}^\sqrt{}$ if it is zero or not.

If it is impossible to give such a procedure, I would like to know why.