Homology Groups and Ideal Class Groups II—Higher Order Genus Theory

Abstract

AbstractIn this chapter we first study the 2-part of the 1st homology group of a double covering of the 3-sphere ramified over a link, introducing the higher order linking matrices which are defined by using the Milnor numbers of the link in Chap. 8. Imitating the method for a link, we study the 2-part of the narrow ideal class group of a quadratic extension of the rationals, using the arithmetic Milnor numbers introduced in Chap. 8. Our theorem may be regarded as a higher order generalization of Gauss’ and Rédei’s theorems on the 2-rank and 4-rank of the ideal class group.KeywordsNarrow Ideal Class GroupHomology GroupsMilnor NumberHigher Order GeneralizationQuadratic ExtensionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.