Multiplicative groups in some infinite algebraic number fields
An abelian group A is called regular when the factor group of A modulo the torsion subgroup of A is a free abelian group. This paper proves that if F is an algebraic number field whose Galois closure contains only a finite number of roots of unity, then the multiplicative group of F is regular. In particular, the multiplicative group of every totally real algebraic number field becomes regular. It also follows that the unit group of every CM-field is regular. On the other hand, the multiplicative group of the maximal CM-field turns out to be isomorphic to a direct sum of a regular group and countably many infinite copies of the additive group of the rational numbers. Incidentally, it is proved that for any prime number p, if K is a CM- field with all roots of unity of p-power orders, then the Galois group of the maximal p-extension of K over K is isomorphic to a free pro-p-group with countably many free generators.
multiplicative group of algebraic number field
scientific article; zbMATH DE number 4077374