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Assuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number field F with the orders of the tame kernels of some of its subfields. The details are given for fields F which are Galois over ℚ with Galois group the group ℤ/2 × ℤ/2, the dihedral group D2p; p an odd prime, or the alternating group A4. We include numerical results illustrating these formulas.

We provide the reader with a uniform approach for obtaining various useful explicit upper bounds on residues of Dedekind zeta functions of numbers fields and on absolute values of values at $s=1$ of $L$-series associated with primitive characters on ray class groups of number fields. To make it quite clear to the reader how useful such bounds are when dealing with class number problems for CM-fields, we deduce an upper bound for the root discriminants of the normal CM-fields with (relative) class number one.

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Tame kernels and second regulators of number fields and their subfields

Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of L-Functions at s = 1, and Explicit Lower Bounds for Relative Class Numbers of CM-Fields

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Tame kernels and second regulators of number fields and their subfields

Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of L-Functions at s = 1, and Explicit Lower Bounds for Relative Class Numbers of CM-Fields