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On duality for skew field extensions. (English) Zbl 0661.16015
Let N be a skew field and \(L\supseteq K\), \(L_ 1\supseteq K_ 1\) subfields (possibly skew). The extensions L/K and \(L_ 1/K_ 1\) are said to be dual if L, K have centralizers \(K_ 1\), \(L_ 1\) and \(L_ 1\), \(K_ 1\) have centralizers K, L in N. In this paper the author makes a study of such dual extensions. He shows (using field coproducts) that any extension L/K can be embedded in a field N in which it has a dual \(L_ 1/K_ 1\) and the right degree of L/K equals the left degree of \(L_ 1/K_ 1\). Moreover dual extensions have dual lattices of intermediate fields. The author considers the following four types, where \(Z_ L(K)\) denotes the centralizer of K in L: L/K is central if \(K.Z_ L(K)=L\), plain if \(K.Z_ L(K)=K\), outer if \(Z_ LZ_ L(K)=L\) and inner if \(Z_ LZ_ L(K)=K\). An extension is plain if and only if its dual is outer; if L/K is central, its dual is inner, and the converse holds provided L/K has finite left degree. An element c of L is said to be normalizing for K if \(c\neq 0\) and \(cK=Kc\); if L has a normalizing basis over K, the extension is said to be normalizing. Now for dual extensions of finite degree, L/K is Galois if and only if \(L_ 1/K_ 1\) is normalizing. The author examines a number of special cases, thus he shows that \(L_ 1/K_ 1\) is a crossed product if and only if K is the fixed field of the group G of L/K and \(| G| =[L:K]\).
Reviewer: P.M.Cohn

MSC:
16Kxx Division rings and semisimple Artin rings
16S20 Centralizing and normalizing extensions
12E15 Skew fields, division rings
16W20 Automorphisms and endomorphisms
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