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We consider the problem of whether the norm one torus defined by a finite separable field extension K/k is stably (or retract) rational over k. This has already been solved for the case where K/k is a Galois extension. In this paper, we solve the problem for the case where K/k is a non-Galois extension such that the Galois group of the Galois closure of K/k is nilpotent or metacyclic.

[Abstract] The authors present interference analysis toward the FS (Fixed Service) communication systems by the HEO (Highly Elliptical Orbit) satellite systems in Ka-band (17.7 – 19.2GHz). Especially, we treat two kind of HEO systems such as “Geosynchronous HEO” and “Sub-geosynchronous HEO” defined in ITU-R Recommendation S.1758. Geosynchronous HEO consists of 3 satellites of 24-hour period. Sub-geosynchronous HEO uses 3 satellites of 12-hour period. The authors provide the interference distribution maps and analyze the detail interfering conditions, by calculating I/N value on the assumption that each satellite system radiates interfering signals for 24 hours. The calculated I/N results are compared with the interference protection criteria determined by ITU-R Recommendation F.1495. Geosynchronous HEO satisfies the criteria perfectly because this is the regional communication system and transmit PFD toward the low elevation angle area is very small. On the other hand, Sub-geosynchronous HEO can not satisfy the long term protection criteria even if this system uses the modified PFD mask which is 8dB less than the RR mask in low elevation area. It is necessary to reduce I/N not only by suppressing PFD but also by using other methods.

Let K be a Dedekind domain with quotient field K, and let A be an H order in a separable K-algebra A. A. Frohlich [4] introduced the locally free Picard group of A, which is denoted by LFP(,4). This is defined to be the group of all isomorphism types of invertible /i-/i-bimodules in A which are locally free as left n-modules. As usual, the locally free class group of ,4 is denoted by C(A). It is noted that if/i is commutative, then LFP(/I) z C(/i). Let Aut(/l) be the group of all automorphisms of n and let In(A) be the subgroup of Aut(/i) consisting of all inner automorphisms of/i. Let cent@) denote the center of /i. An automorphism f of /i is called a central automorphism iff(c) = c for any c C ccnt(/i). WC denote by Autccnt(/l) the subgroup of Aut(/l) consisting of all central automorphisms of A, and define Outcent = Autcent(/i)/In(/l). Fundamental properties of the groups LFP(A) and Outcent were systematically studied by Friihlich, Reiner and Ullom (4, 5 1. Let G be a finite group and let LG be its integral group ring. We denote by C, the cyclic group of order n, by D, the dihedral group of order 2n and by H, the quaternion group of order 4n. Frohlich, Reiner and Ullom computed LFP(ZG) and Outccnt(LG) when G = D,, II, or D,, p an odd prime. 286 0021.8693/82/08028~25%02.00/0

Compensation techniques for rain depolarization in satellite links are discussed from the viewpoint of system configuration, performance, and technical feasibility. The compensator using two rotatable polarizers has sufficient performance at microwave region. Two types of compensators, as combination of 90° and 180° polarizers and a combination of two 90° polarizers, were developed and tested. Satisfactory results were obtained at 4-GHz band by using the INTELSAT satellite. The compensation system for an operational use is also presented in this paper.

From the viewpoint of effective utilization of both the geostationary orbit and the frequency spectrum in the satellite communication system, it is strongly required of an earth station antenna to have low sidelobe characteristics in order to minimize the interference from/to the adjacent satellite and the terrestrial microwave links. The reference diagram of the sidelobe gain for a large antenna recommnended by International Radio Consultative Committee (CCIR) is drawn as follows: G = 32-25 \log \theta dBi ( 1 \deg \leq \theta \leq 48\deg ); G = -10 dBi ( \theta > 48\deg ), where G is the sidelobe gain relative to an isotropic antenna and \theta is an offset angle from the main lobe axis in degrees. In the case of an axisymmetrical Cassegrain antenna operating with circular polarization, the level of the cross-polar sidelobe due to diffraction by the subreflector support struts exceeds that shown in the above reference diagram. By introduction of a new type of strut, the sidelobe gain can be suppressed below that shown in the CCIR reference diagram. In this article, a design method based on geometrical theory of diffraction (GTD) for improving the strut shape is presented. The wide-angle sidelobe gain of the standard " A " earth station antenna, installed in Yamaguchi, Japan, being equipped with the improved strut, can meet that shown in the CCIR reference diagram in all directions for both co- and cross-polar polarizations in the 6 GHz band.

Let A be a Z-order in a semisimple Q-algebra -4. Let C(A) denote the class group of A defined by using locally free left A-modules. Define D(d) to be the kernel of the natural map C(A) + C(Q) for Q a maximal Z-order in A containing A and d(A) to be the order of D(d). The integral group ring ZG of a finite group G is a Z-order in the semisimple Q-algebra QG, and hence the groups C(ZG) and D(ZG) can be defined. The study of the groups C(ZG) and D(ZG) seems to be important, because it has applications to various problems in number theory, algebraic topology, etc. In this paper we will try, as a continuation of [3], to determine finite nonabelian groups G such that d(ZG) = 1. Denote by D, the dihedral group of order 2n and by S, , A, the symmetric, alternating group on n symbols, respectively. The following theorem has been obtained in [3].

A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

Let R be a commutative ring and K be the total quotient ring of R. Let Σ be a separable K-algebra which is a finitely generated projective, faithful K-module and Λ be an R-order in DΛ/R. We denote by DΛ/R the Dedekind different of Λ and by NΛ/R the Noetherian different of Λ.

Publisher Summary This chapter discusses the rings of invariants of finite linear groups. Let τ be a field. An element τ of finite order in GLn(k) is called a pseudo-reflection if rank(τ − I) = 1. A finite subgroup of GLn(k) is called small if it contains no pseudo-reflection. The chapter discusses a few basic properties of the fundamental groups of Noetherian normal domains. The chapter presents the assumption that R is a Noetherian normal domain of which k is the quotient field and Ks is the fixed separable closure of k. Also, L is a finite extension of K, and S is the integral closure of R in L. Then, S is a 1-unramified extension of R if every prime ideal of height 1 in S is unramified over R.

Let Π be a finite group and denote by MΠ the class of finitely generated Z-free ZΠ-modules. In [2] we defined a certain equivalence relation on MΠ and constructed the abelian semigroup T(Π), which was studied in [3] (see [1] and [5], too).

In this note we assume that all rings have identities and denote by R a commutative ring. All R-algebras considered are assumed to be finitely generated faithful R-modules. An R-algebra Λ is said to be semi-simple ([5]), if any finitely ‘generated Λ-module is (Λ, R)-projective.

Recently A. Hattori introduced in [7], [8], [9] the notion of simple algebras over a commutative ring. Especially, in [9], he examined, as a fundamental problem on simple algebras, whether a directly indecomposable semi-simple algebra is simple or not and gave affirmative answers to this in some particular cases. In this note we shall first prove, as a complete answer to this, that any directly indecomposable semi-simple algebra over a Noetherian ring is simple.

The ramification theory in commutative rings, as a generalization of the classical one in maximal orders over a Dedekind domain, was established in [1], [13], [15] and etc.. For non-commutative algebras this was also studied in [3], [4], [9], [18] and etc.. However, the different theorem, which is a central part of ramification theory, has not been given in those, except for some special cases (cf. [12], [18]). The main object of this paper is to give the discriminant theorem and the different theorem for projective orders in a (non-commutative) separable algebra, in the most general form.