The call type(expr, radalgnum) checks if expr is an algebraic number in RootOf or radical notation.

An algebraic number is defined as either a rational number, a root of a univariate polynomial with algebraic number coefficients, specified by a RootOf, or an n-th root of an algebraic number. A sum, product, or quotient of these is also an algebraic number.

For example, both $\sqrt{2}$  and RootOf(z^2-2, z) are of type radalgnum.

$\mathrm{type}\left(\frac{2}{3}\,\mathrm{radalgnum}\right)$

$\mathrm{true}$

$\mathrm{type}\left(\mathrm{RootOf}\left(z^2+1\,z\right)\,\mathrm{radalgnum}\right)$

$\mathrm{true}$

$\mathrm{r1}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^5+\mathrm{\_Z}+2^{\frac{1}{2}}\right)$

$\mathrm{r1}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^5+\mathrm{\_Z}+\sqrt{2}\right)$

$\mathrm{type}\left(\mathrm{r1}\,\mathrm{radalgnum}\right)$

$\mathrm{true}$

$\mathrm{type}\left(\mathrm{r1}\,\mathrm{algnum}\right)$

$\mathrm{false}$

$\mathrm{r2}≔{\left(2+\mathrm{RootOf}\left({\mathrm{\_Z}}^5+\mathrm{\_Z}+3\right)\right)}^{\frac{2}{3}}$

$\mathrm{r2}≔{\left(2+\mathrm{RootOf}\left({\mathrm{\_Z}}^5+\mathrm{\_Z}+3\right)\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{type}\left(\mathrm{r2}\,\mathrm{radalgnum}\right)$

$\mathrm{true}$

$\mathrm{type}\left(\mathrm{r2}\,\mathrm{radnum}\right)$

$\mathrm{false}$

$\mathrm{type}\left(\mathrm{RootOf}\left({\mathrm{\_Z}}^{\frac{1}{2}}+2\right)\,\mathrm{radalgnum}\right)$

$\mathrm{false}$

$\mathrm{type}\left(\ln \left(2\right)\,\mathrm{radalgnum}\right)$

$\mathrm{false}$

$\mathrm{type}\left(\frac{5}{\mathrm{RootOf}\left(z^2-2\,z\right)}\,\mathrm{radalgnum}\right)$

$\mathrm{true}$

$\mathrm{type}\left(\mathrm{RootOf}\left(z^2+y\,z\right)\,\mathrm{radalgnum}\right)$

$\mathrm{false}$