Basic noncommutative Galois theory
Let \(K \subseteq L\) be a pair of skewfields such that for every \(\ell \in L - K\) there exists some automorphism \(g:L\to L\) of \(L\) with \(g(k) = k\) for all \(k \in K\), but \(g (\ell) \neq \ell\). In this case \(L\) is called a Galois extension of the skewfield \(K\), while the group \(\text{Aut}_K(L)\), consisting of all automorphisms of \(L\) which keep every element in \(K\) fixed, will be called the Galois group of that Galois extension. A Galois extension \(L\) of a skewfield \(K\) is said to be finite if the dimension \(\dim_K (L_K)\) of \(L\), considered as a right \(K\)-vector space, is finite. Using the Wedderburn-Artin theory of simple artinian rings, the author gives a proof for the following well-known theorem: For every finite Galois extension \(L\) of a skewfield \(K\) with Galois group \(G\) there exists a canonical one-to-one correspondence between (a) subskewfields \(F\) of \(L\) containing \(K\), on the one hand, and (b) those subgroups \(U\) of \(G\), on the other hand, which satisfy the following additivity condition: (Add) If \(\ell_1\), \(\ell_2\), and \(\ell_1 + \ell_2\) are nonzero elements in \(L\) and if both conjugation by \(\ell_1\) and conjugation by \(\ell_2\) are contained in \(U\), then conjugation by \(\ell_1 + \ell_2\) is also contained in \(U\).
scientific article; zbMATH DE number 745279