Show that if \(L/K/F\) with \(K/F\) algebraic and \(L/K\) algebraic then \(L\) is algebraic.
Solution
Let \(\alpha\in L\). Since \(L/K\) is algebraic, \(\alpha\) satisfies a nonzero polynomial \[ p(x)=a_0+a_1x+\cdots+a_nx^n\in K[x]. \] Because \(K/F\) is algebraic, the finitely many coefficients \(a_i\) are algebraic over \(F\). Hence \[ F_0:=F(a_0,\dots,a_n) \] is a finite extension of \(F\).
Now \(p\in F_0[x]\), so \(\alpha\) is algebraic over \(F_0\). Therefore \(F_0(\alpha)/F_0\) is finite. By the tower law, \[ [F_0(\alpha):F]=[F_0(\alpha):F_0][F_0:F]<\infty. \] Thus \(\alpha\) is algebraic over \(F\). Since \(\alpha\in L\) was arbitrary, \(L/F\) is algebraic.