A simple, (almost) non-inductive proof for the existence of composite fields and algebraic closures (Q1368000)

The author gives a non-inductive proof for the existence of composite fields and algebraic closures. Let \(L_1/K\) and \(L_2/K\) be field extensions. Then there exists a field extension \(L/K\) and \(K\)-embeddings \(\phi_1: L_1\to L\) and \(\phi_2: L_2\to L\) such that \(L\) is generated by \(\phi_1(L_1)\) and \(\phi_2(L_2)\) over \(K\). \(L\) is said to be the composite of \(L_1\) and \(L_2\). The author proceeds as follows. One considers the \(K\)-algebra \(R'= \text{End}_K(V)\) of \(K\)-endomorphisms of the \(K\)-vector space \(V= \text{Hom}_K(L_1, L_2)\) into which \(L_1\) and \(L_2\) can be embedded. Let \(R\) denote the subring generated by the images of \(L_1\) and \(L_2\) of these embeddings. Then one may choose a maximal ideal \(M\) of \(R\) and let \(L= R/M\). More general approaches are also discussed.