The command IsAlgebraic(v, rc, R) returns true if v is algebraic with respect to rc, false otherwise.

This command is part of the RegularChains[ChainTools] package, so it can be used in the form IsAlgebraic(..) only after executing the command with(RegularChains[ChainTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][IsAlgebraic](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\,t\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$T≔\mathrm{Empty}\left(R\right)\:$

$T≔\mathrm{Chain}\left(\left[\left(z+1\right)\left(z+2\right)\,y^2+z\,\left(x-z\right)\left(x-y\right)\right]\,T\,R\right)$

$T≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(T\,R\right)$

$\left[x^2+\left(-y-z\right)x+zy\,y^2+z\,z^2+3z+2\right]$

$\mathrm{IsAlgebraic}\left(x\,T\,R\right)$

$\mathrm{true}$

$\mathrm{IsAlgebraic}\left(t\,T\,R\right)$

$\mathrm{false}$