Generators and Relations for O

Abstract

Tate's algorithm for computing K 2 O F for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order-the latter, together with some structural results on the p-primary part of K 2 O F due to Tate and Keune, gives a proof of its structure for many number fields of small discriminants, confirming earlier conjectural results. For the first time, tame kernels of non-Galois fields are obtained.

Teichmüller spaces for pointed Fuchsian groups

“…These equations are expected to be given a proof. Using the computing programm in [1], we compute that all the K-groups are confirmed with the conjectural order and give their structures.…”

“…Using method in [1], we have K 2 (O F ) is isomorphic to Z/2 × Z/2. Moreover, we get the equation as follows by the numerical method…”

“…Using PARI, we have α ≈ 1.007552359378179 + 0.513115795597015i D(α 2 ) ≈ 0.981368828892232 ζ F (2) ≈ 1.056940574599707 d F = −283Assuming Lichtenbaum conjecture and [α 2 ] being the base of B(F ), since w 2 (F ) = 24, we have#K 2 (O F ) = 3By PARI, we have #K 2 (O F ) = 3.99999999999999 = 4. Using method in[1], we have K 2 (O F ) is isomorphic to Z/2 × Z/2 and the function equation is ζ F (2) = 16π (α 2 ).…”

“…By PARI we have #K 2 (O F ) = 4.000000000000000 = 4. Using method in[1], we have K 2 (O F ) is isomorphic to Z/2 × Z/2. The equation of zeta function at 2 is…”

Topics in computational algebraic number theory

“…Note that p/p is ramified if and only if p 2 ∩ Z = (pZ) in Step (2b). If p/p is unramified, then we can take π = p in Step (4); otherwise, at least one of the γ i does not belong to p 2 , and this is easy to test since the HNF for p 2 is known. The reduction modulo p in the last step ensures that a small element is returned, and the test π ∈ p 2 is only needed when e(p/p) = 1.…”