Eisenstein führte in seiner Abhandlung [1] einen durchsichtigen, analytischen Beweis des Reziprozitätsgesetzes der Potenzreste im rationalen und Gaussschen Zahlkörper. Zur kurzen Fassung seiner Methode im rationalen Fall, seien m und n positive ganzrationale Zahlen und sei

.

Es gibt zwei verschiedene Arten von Summen, die gewöhnlich Gausssche Summen genannt werden. Die eine Summe wird von einer Zahl ω ≠ 0 eines algebraischen Zahlkörpers F bestimmt.

For convenience we consider throughout an arbitrary but fixed multiplicative group H. The integral group ring of H is denoted by ZH, and the homomorphism ε: ZH→Z is always the trivializer, or unit augmentation, defined by εh = 1 for all h ∈ H.

The concept of a “set of ρ-discs” for complex-valued functions defined in the open unit disc U was introduced in [4; 5, p. 265] and various results concerning the existence of such sets associated with spirals and Stolz angles were derived. These sets are non-Euclidean analogues to the classical cercles de remplissage of Milloux [7; 8]. In the present paper additional theorems about Stolz angle phenomena and the existence of ρ-discs in more general subsets of U are established and new proofs are indicated for some results of Seidel [9, Theorem 2 and Corollary 1].

Let L > K ≧ Φ, LK, be three fields such that: (1) L/K is not purely inseparable, and (2) L/Φ is transcendental. Then Herstein’s theorem [2] asserts the existence of u ∈ L such that f(u) ∉ K for every non-constant polynomial f(X) ∈ Φ[X].

In this chapter, we shall formulate, without proof, the theorys of exterior differential systems and of their prolongations using the language in the theory of jets developed by C. Ehresman, because such formulation seems to be most convenient in order to apply the theory to the theory of continuous infinite pseudo-groups, which we shall discuss in the next chapter. Since important theorems in our theory hold only in the real analytic case, we shall exclusively consider the real analytic case. So, we shall omit the adjective “analytic”, unless explicitly stated otherwise. As for the fundamental notions in the theory of jets and of differential systems, we refer to [5] and [6], respectively. Detailed proof of the contents of this Chapter will be published in [7].

The object of the present paper is to prove some theorems concerning the existence and the uniqueness of the solution of the initial value problem for the evolution equation

(E).

In Part I we introduced the notion of 2. Frobenius extensions of a ring, as a generalization of Kasch’s [10] Frobenius extensions and hence of classical Frobenius algebras. We proved, in I, bilinear (or sesqui-linear, rather, to follow Bourbaki’s terminology) form and scalar product characterizations of Frobenius extensions in such extended sense, generalizing Kasch’s and classical case, and then studied homological dimensions in them, generalizing and refining the results in Eilenberg-Nakayama [4] and Hirata [6]. Dual bases were considered in case of quasi-free (2.) Frobenius extensions Also the case of a semi-primary or S-ring ground ring was studied.

Recently S. S. Chern [1] intended an aproach to some problems about analytic mappings of Riemann surfaces from a view-point of differential geometry. In that line we treat here orders of circular points of analytic mappings. The author expresses his thanks to Prof. K. Noshiro for his kind advices.

Our purpose in this note is to establish a classification theorem for fiberings with a loop-space as fibre. This is deduced by applying a generalized suspension theorem which will be proved in § 1. As a by-product we obtain a proposition concerning fiberings with a loop-space as the total. Throughout this note we shall denote by the category of spaces having the based homotopy type of a CW-complex.

Let k be an algebraic number field of finite degree, A the maximal abelian extension over k, and M a meta-abelian field over h of finite degree, that is, M/k be a normal extension over k of finite degree with an abelian group as commutator group of its Galois group.

Let D be a plane domain containing the point at infinity and E its complementary closed set. As to a sufficient condition for a compact set E to be of class N, Pfluger-Mori’s criterion is well-known (Pfluger [10], Mori [6]). Various relations between the conditions of this type and the Hausdorff measure of the set E have been investigated recently by Kuroda and Ozawa (Kuroda [5], Ozawa and Kuroda [8], Ozawa [7]). For example they showed that Pfluger-Mori’s condition implies that the set E is of one dimensional measure zero under some additional conditions (cf. [7], [8]). In the present paper we shall give an alternative proof of Pfluger-Mori’s criterion and another criterion using analytic module and, further, prove some criteria for the set E to be of one dimensional measure zero.

This note is a correction of an error of the author’s paper mentioned in the title. (The reference [1]). The proof of the Prop. 6 of [1], Chap. II, p. 247, is an error. And the propositions and formulas in pp. 247-249 of [1] depending on this Prop. 6 must be corrected. All notations are referred to [1].