Automorphism groups of fields
This paper continues a long series of publications of Kuyk, Fried and Kollar, Geyer, Pröhle, Stichtenoth, Dugas and Göbel, Schmelzer and others. By two different constructions the following theorem is extended. To every group \(G\) and every field \(K\) there exists a field extension \(F\) with \(\Aut F\simeq G\) and \(K \subseteq \text{Fix} (\Aut F)\) where \(\text{Fix} (\Aut F)\) denotes the fixed field of \(\Aut F\). Especially \(K= \text{Fix} (\Aut F)\) holds, if \(G\) is infinite. 1) If \(K\) does not contain too many roots of unity then the cardinality of \(F\) can be estimated by \(| F|\leq \max \{\aleph_ 0, | G|, | K|\}\). 2) The theory of fields is ``group universal''. Let a group \(G\), a field \(K\) and a homomorphism \(\varphi: G\to \Aut K\) be given. Then there exists a field extension \(F\) with \(\Aut F\simeq G\) and \(\varphi (G)\) is the restriction of \(\Aut F\) on \(K\). If one restricts to right orderable groups the theory of formal real fields is group universal, too. The question for group universality was raised by Funk, Kegel and Strambach who showed this property for other algebraic structures with quite different methods. As in other papers the authors consider a transcendental base \(X\) on which the group \(G\) operates faithfully. The field \(K(X)\) can be extended algebraically to \(F\subseteq \widehat {K (X)}\) such that \(X\) is mapped onto itself by \(\Aut F\). Further \(\Aut F\) operates on \(X\) in the same way as \(G\). The elements of \(X\) are distinguished by root chains. To show group universality the authors have adjusted their construction from 1987 [Trans. Am. Math. Soc. 304, 355-384 (1987; Zbl 0644.12011)] which uses combinatorial tools from S. Shelah. From there they get a cardinal jump for \(F\) so that it remains open whether 1) and 2) hold simultaneously. As in their paper (loc. cit.) monoids with right cancellation law were considered instead of groups, that paper also presented an essential extension of the theorem quoted in the beginning. As the reviewer has shown in his paper [Result. Math. 25, No. 3-4, 357-369 (1994; Zbl 0815.12003)] 1) holds without any restriction on the ground field \(K\). Furthermore it will be interesting to know in which respect 1) or 2) hold for endomorphism monoids.
scientific article; zbMATH DE number 724655
scientific article