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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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##### References:
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