Universiteit Leiden

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Proefschrift

The unit residue group

The unit residue group, to which the present thesis is devoted, is defined using the norm-residue symbol, which Hilbert introduced into algebraic number theory in 1897.

Auteur
Dalla, Torre G.
Datum
18 december 2019
Links
Thesis in Leiden Repository

The unit residue group, to which the present thesis is devoted, is defined using the norm-residue symbol, which Hilbert introduced into algebraic number theory in 1897. By its definition, the unit residue group of a global field is a direct sum of local contributions. It has a subgroup of a global nature, called the virtual group.We give a precise description of the unit residue groups and their virtual subgroups for some classes of number fields, including all quadratic fields. In addition we point out connections to two classical theorems on ideal class groups, namely the theorem of Armitage and Froehlich on 2-ranks and Scholz’s theorem on 3-ranks.We also study certain subgroups of the multiplicative group of a local field that play an important role in an algorithm for computing norm-residue symbols, and group isomorphisms between the groups of quadratic characters of two number fields that preserve L-series.

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