Tame kernels and second regulators of number fields and their subfields

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.

VALUES AT s = -1 OF L-FUNCTIONS FOR MULTI-QUADRATIC EXTENSIONS OF NUMBER FIELDS, AND THE FITTING IDEAL OF THE TAME KERNEL

Fix a Galois extension [Formula: see text] of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in [Formula: see text], let [Formula: see text] denote the primes of [Formula: see text] lying above those in S, and let [Formula: see text] denote the ring of [Formula: see text]-integers of [Formula: see text]. We then compare the Fitting ideal of [Formula: see text] as a ℤ[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of ℚ[G], and hence in ℤ[1/2][G]. Results in ℤ[G] are obtained under the assumption of the Birch–Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where [Formula: see text] is a biquadratic extension of F containing the first layer of the cyclotomic ℤ2-extension of F, and describe a class of biquadratic extensions of F = ℚ that satisfy this condition.