One more shortcut to Galois theory (Q1842221)

The author gives a short proof of the basic structure theorem of Galois theory using nothing but Dedekind's lemma, one of its elementary and well-known consequences, and two of the most basic facts of \(G\)-set theory. The same direct approach is used by the author to establish for a finite group \(G\) of automorphisms of a field \(L\) with fixed field \(K\) the canonical (anti-) equivalence of the category of finite-dimensional \(L\)- split \(K\)-algebras and the category of finite \(G\)-sets as well as the basic existence theorem of Galois theory, that is, the fact that the number of \(K\)-algebra automorphisms of a field extension \(L = K(\alpha_1, \dots, \alpha_n)\) with \(\#\{\alpha_1, \dots,\alpha_n\} = n\) and \(\prod^n_{i = 1} (X - \alpha_i) \in K[X]\) equals the degree \((L:K) = \text{Dim}_K L\) of this extension and is bounded above by \(n!\).