Zahlentheoretische S�tze aus der Gruppentheorie

Abstract

In c~en folgendea Zeilen wird ein System von Matrizen behandelt, das nach einem, yon dem gewShnlichen verschiedenen Gesetz der Matrizen-komposi~ion eine Gruppe bildet. Satz 1 gibt volls~ndigen &ufschlul~ iiber die Natur derartiger Gruppen. w 2 enthMt Anwendungen auf die Frage, welches der einfachste algebraische KSrper ist, in dem eirre Darstellung einer endlichen Gruppe yon Matrizea (ira gewShnlichen Sinne) gegeben werden karma). Wit zeigen u. a., daL~ dies fiir eine irreduzible Darstellung ungeraden Grades mit reellem Charakterensystem stets der KSrper der Charaktere ist. w Gegeben ski ein algebraischer KSrper K, der in dem GrundkSrper k Galoissch ist und in diesem Bereieh die Gruppe ~ mit den Elementen E,

Finite linear groups of degree seven. II

“…Here η is the nontrivial character of Z 2 , Z 2 is the group of order 2. As X is rational and of odd degree it can be written in the real field [22]. We may assume the matrices are orthogonal.…”

Rational functions invariant under a finite abelian group

“…Remark that 1(Μ)π is rational over Γ if and only if a certam torus, defmed over Γ and Splitting over /, is rational over Γ, cf [38] This will not be used m the sequel We usually wnte the group law in M additively, although M is a sub-7r-module of the multiphcative group of i (M) (l 3) Proposition [43] Lei W be an l-vector space on which π acts semilmearly, ι e W is a π-module and σ(λ\ν) = (σλ) (σ w) for all σεπ, Ae/ and weW Then WK contams an l-basis for W Prooj Put S = (£ σ)εΖ [π] We show that SWc Wn contams an /-σεπ basis by provmg that any /-linear function φ W~^l annihilatmg SW must be the zero function Fix such a. φ, and fix weW Ther for every Ae/ we have By the linear mdependence of field automorphisms [2, Ch V, § 7 5] we conclude φ (σ w) = 0 for all σ e n In particular φ ( w) = 0, and ( l 3) follows Q (l 4) Proposition [30] Let N be a fmitely generated permutation module over π Then 1(Ν)π is rational over Γ Proof Let {xl , ,\}c~-l(N)* be a Z-basis for N which is permuted by π is an exact sequence of Jinitely generated Z-free π-modules, then 1(Μ2)π is rational over Proof. From (1.5) we get /(Μ2)π^/(Μ1ΘΛΓ)π, and (1.4), applied to the base field /(Mt) instead of /, says that /(Λ^φΛ/71 is rational over l(Mj)".…”

Cohomology and Galois theory. I. Normality of algebras and Teichmüller’s cocycle

“…Conversely, Speiser's "Principal genus theorem in minimalen" ( [12], cf. [2, p. 57]) asserts that every function/ with 5/=l has the form/ = 5c, for some constant c in N.…”