Let $p\in \mathbb C[x_1,\dots,x_n]$, and $B,C\subset\mathbb C[x_1,\dots,x_n] $ two subsets. I am trying to show by hand that if $p$ is algebraic over $B$ and $B$ algebraic over $C$, then $p$ is algebraic over $C$.
Any idea or reference ?
Edit. Here $B$ algebraic over $C$ means that every $b\in B$ is algebraic over some $c_1,\dots,c_n$ in $C$, which means that there are polynomials $Q_0(\bar x),Q_1(\bar x),\dots,Q_m(\bar x)\in \mathbb C[x_1,\dots,x_n]$, such that $Q_1(c_1,\dots,c_n)b+\dots+Q_m(c_1,\dots,c_n)b^m=Q_0(c_1,\dots,c_n)$ and $Q_m(c_1,\dots,c_n)\neq 0$.