Let \(F \subseteq K \subseteq L\) be a tower of field extensions. Prove that if \(K/F\) is algebraic and \(L/K\) is algebraic, then \(L/F\) is algebraic.
Solution
Let \(\alpha\in L\). Since \(L/K\) is algebraic, \(\alpha\) satisfies a nonzero polynomial \[ p(x)=c_0+c_1x+\cdots+c_nx^n\in K[x]. \] Let \[ E=F(c_0,\dots,c_n)\subseteq K. \] Because \(K/F\) is algebraic, each \(c_i\) is algebraic over \(F\). A field generated by finitely many algebraic elements is finite over the base field, so \[ [E:F]<\infty. \] Since \(p\in E[x]\) and \(p(\alpha)=0\), the extension \(E(\alpha)/E\) is finite. Hence by the tower law, \[ [E(\alpha):F] =[E(\alpha):E][E:F]<\infty. \] Every finite extension is algebraic, so \(\alpha\) is algebraic over \(F\). Since \(\alpha\in L\) was arbitrary, \(L/F\) is algebraic.