Seems a naive question, but I get stuck there..
Say $M/L/K$ is a tower of fields, $L/K$ is algebraic, $B\subset M$. ($B$ is not necessarily algebraic independent over $K$ or $L$.) Is $L(B)/K(B)$ also algebraic?
My thought is as follows: take $h(b_1,...,b_n)=\frac{f(b_1,..,b_n)}{g(b_1,...,b_n)}\in L(B)$, say all coefficients form a finite subset $C\subset L$.Then $f,g\in K(C)[B]$. Clearly $[K(C):K]<\infty$. If we can show $[K(C,B):K(B)]<\infty$, then we are done. And we can see elements in K(C)[B] are clearly linear combinations of basis elements of K(C)/K with coefficients in K[B]. But how can we carry over to the field of fractions?
I see that's really a naive question.. Being algebraic is preserved under division.. Thanks for everybody's attention..