Number Theory and its Development to Discrete Mathematics

围绕这一研究主题的核心课题：数论，特别是与阿贝尔域[A]、算术、代数几何[B]及其在离散数学[C]中的应用有关的Hasse问题，我们在2008~2010年的每年8月和1月在佐贺举办了数论研讨会。研究组织者在两年半的时间里留在了NUCES，与两个校区的博士学者共同开展研究。关于Hasse的问题，组织者通过与NUCES的博士学者的合作，获得了关于纯六次和纯八次场族的单一性的刻画。在我们对代数几何码的研究中，Uehara合作给出了构造一类不同于已知的一点型码的代数几何码的想法。此外，上原还应用了推广一点型代数几何码的评估码的概念，发明了一种由代数数域上的整数环构造码的方法，并提出了S…这类代码的更多显式例子[C]。合作宫崎骏研究了Buchsbaum变种的最小自由分辨，得到了Buchsbaum变种的Castelnuovo-Mumford正则性分类[B]。合作的Terai研究了Stanly-Reisner理想，它是多项式环上的无平方单项理想[B]。片山合作社。利用三次和二次域的单位群的结构确定了立方和四阶有限辛群，并在2011年的Saga研讨会上宣布了这些结果。Newman，Shanks和Williams在1980年的S中确定了有限平方阶辛群。片山还研究了单位圆内接的全余k-多边形数，其中的顶点选自该圆的n个分割点[C]。田口合作研究了截断离散赋值环(=：tdvr‘s)的分支理论和mod p Galois表示的(非)存在性。在前者，Taguchi(与T.Hiranouchi)证明了Tdvr A的有限扩张范畴等价于提升A的完备离散赋值域的有限扩张范畴[B]。在后者，Taguchi(与H.Moon)证明了某些二次域不存在二维mod2 Galois表示[B]。较少

On the core subjects of this research theme ; Number Theory, specifically Hasse's problem related to Abelian fields[A], Arithmetic, Algebraic geometry[B] and its application to Discrete Mathematics[C], we held the Workshop on Number Theory in Saga in each August and January of 2008~2010. The research organizer stayed at NUCES during two and half years to work the joint research with PhD scholars at both campuses. On Hasse's problem, the organizer obtained the characterization on the monogeneity for certain family of pure sextic and pure octic fields by the joint work with PhD scholars at NUCES. In our research on algebraic geometry codes, Cooperative Uehara gave an idea of constructing a new class of algebraic geometry codes different from the known codes of one-point type. Also, applying the concept of evaluation codes, which generalize one-point-type algebraic geometry codes, Uehara invented a method of constructing codes from integer rings on algebraic number fields, and presented s … More ome explicit examples of such codes[C]. Cooperative Miyazaki studied the minimal free resolution of Buchsbaum varieties and obtained a classification of the Buchsbaum variety in terms of the Castelnuovo-Mumford regularity[B]. Cooperative Terai studied Stanly-Reisner ideals, which are squarefree monomial ideals in polynomial rings[B]. Cooperative Katayama . has determined finite symplectic groups of cube and 4th order, using the structure of the unit groups of cubic and quadratic fields, and announced these results at the workshop in Saga 2011. Newman, Shanks and Williams determined finitesymplectic groups of square order in1980's. Katayama also investigated the number of congruent k-polygons inscribed in a unit circle, where the vertices chosen from n division points of the circle[C]. Cooperative Taguchi studied the ramification theory of truncated discrete valuation rings (= : tdvr's) and the (non)existence of mod p Galois representations. On the former, Taguchi proved (jointly with T. Hiranouchi) that the category of finite extensions of a tdvr A is equivalent to a category of finite extensions, with restricted ramification, of a complete discrete valuation field which lifts A. On the latter, Taguchi proved (jointly with H. Moon) the non-existence of 2-dimensional mod 2 Galois representations for some quadratic fields[B]. Less