an:04196099
Zbl 0725.12006
Kiechle, Hubert
The kernel of an automorphic derivation and an application to normal subfields of generalized Andr??-systems
DE
Arch. Math. 58, No. 5, 514-520 (1992).
00011193
1992
j
12K99
derivation; generalized Andr??-system; kernel; nuclei; fixed field; center of the derived quasifield; normal subfields of nearfields
Let \(Q\) be a quasifield and \(K\) a subfield (not necessary commutative) of the kernel of \(Q\). A map \(\phi: Q^*\to \mathrm{GL}(Q,K)\); \(a\to \phi_ a\) \((Q^*:=Q\setminus \{0\})\) is called a derivation, if the derived quasifield \(Q^{\phi}:=(Q,+,\circ)\) is a quasifield as well. Here \(a\circ b:=a\phi_ a(b)\), \(a\neq 0\) and \(0\circ b:=0\). The subgroup of \(\mathrm{GL}(Q,K)\) generated by \(\phi (Q^*)\) is denoted by \(\Delta_{\phi}\). If \(Q\) is a (skew)field and \(\Delta_{\phi}\) is contained in \(\Aut(Q)\), then \(\phi\) is called automorphic and \(Q^{\phi}\) is usually named a generalized Andr??-system. The kernel of \(\phi\) is the set \(\mathrm{Ker}\,\phi:=\{a\in Q^*;\;\phi_{ax}=\phi_ x\}\), and is a subgroup of \(Q^*\).
This idea implicitly has been used before [\textit{D. A. Foulser}, Math. Z. 100, 380--395 (1967; Zbl 0152.18903); \textit{A. Herzer}, Arch. Math. 52, No. 1, 99--104 (1989; Zbl 0633.51003)]. It proved very useful in the study of the structure of generalized Andr??-systems.
Some relations between \(\mathrm{Ker}\,\phi\) and the nuclei, the fixed field of \(\Delta_{\phi}\) and the center of the derived quasifield are given. These results are used to generalize theorems on normal subfields of nearfields [cf. \textit{H. W??hling}, Theorie der Fastk??rper. Essen: Thales Verlag (1987; Zbl 0669.12014), (III.5.5)]. In the last section some examples are given.
Hubert Kiechle (M??nchen)
Zbl 0633.51003; Zbl 0669.12014; Zbl 0152.18903