This is not a homework assignment or question from an exam but it comes to my mind but I was not able to figure it out. Let $\{u,v\}$ be algebraically independent subset of $\mathbb R$ over $\mathbb Q.$
Put $z:=q v+p$ for some nonzero $p,q\in\mathbb Q$. Clearly, $\{z\}$ is algebraically independent set. My question is how about $$\{u,z\}$$ Is it algebraically independent set as well? Please if you know show me how it prove it or disprove it.
Thank so much in advance.
Note that $\{z\}$ is algebrically independent only if $q\ne 0$. In that case, $\{u,z\}$ is also algebraically independent: Suppose $f(u,z)=0$ with $f\in\Bbb Q[X,Y]$. Then $g(u,v)=0$ with $g(X,Y)=f(X,qY+p)$, hence $g(X,Y)\equiv 0$. But $f(X,Y)=g(X,\frac{Y-p}q)$ (where we make use of $q\ne 0$).