Jerzy Browkin

Tame and wild kernels of quadratic imaginary number fields

For all quadratic imaginary number fields F of discriminant d > −5000, we give the conjectural value of the order of Milnor’s group (the tame kernel) K2OF , where OF is the ring of integers of F. Assuming that the order is correct, we determine the structure of the group K2OF and of its subgroup WF (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, d = −3387).

Tame kernels of cubic cyclic fields

There are many results describing the structure of the tame kernels of algebraic number fields and relating them to the class numbers of appropriate fields. In the present paper we give some explicit results on tame kernels of cubic cyclic fields. Table 1 collects the results of computations of the structure of the tame kernel for all cubic fields with only one ramified prime p, 7 ≤ p < 5,000. In particular, we investigate the structure of the 7-primary and 13-primary parts of the tame kernels. The theoretical tools we develop, based on reflection theorems and singular primary units, enable the determination of the structure even of 7-primary and 13-primary parts of the tame kernels for all fields as above. The results are given in Tables 2 and 3.

Computing the tame kernel of quadratic imaginary fields

J. Tate has determined the group K 2 O F (called the tame kernel) for six quadratic imaginary number fields F = Q(√d), where d = -3,-4,-7, -8,-11, -15. Modifying the method of Tate, H. Qin has done the same for d = -24 and d = -35, and M. Skalba for d = -19 and d = -20. In the present paper we discuss the methods of Qin and Skalba, and we apply our results to the field Q(√-23). In the Appendix at the end of the paper K. Belabas and H. Gangl present the results of their computation of K 2 O F for some other values of d. The results agree with the conjectural structure of K 2 O F given in the paper by Browkin and Gangl.

Continued fractions in local fields, II

The present paper is a continuation of an earlier work by the author. We propose some new definitions of p-adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every m, 1 < m < 5000, 5 m if ∨m E Q 5 Q, then ∨m has a periodic continued fraction expansion. The same is not true in Q p for some larger values of p.

Some new kinds of pseudoprimes

We define some new kinds of pseudoprimes to several bases, which generalize strong pseudoprimes. We call them Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. It turns out that every n < 10 12 , which is a strong pseudoprime to bases 2, 3 and 5, is not a Sylow p-pseudoprime to two of these bases for an appropriate prime p|n-1. We also give examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprimes to two bases only, where p = 2 or 3.

Erratum to “Some new kinds of pseudoprimes”

I am grateful to the editors of Mathematics of Computation for sending me Professor Zhang Zhenxiangs remarks on my paper [1], and am indebted to him for pointing out some inaccuracies. * Page 1032, line 18: Replace from (4) if follows by (4) follows from. * Page 1032, line 24: Replace (4) by (4). * Page 1034, Table 1: In the row labelled n7, shift the data in the last column to one column earlier. * Page 1037, line 18: Replace Elem2 by Syl2. REFERENCES

A theorem of Sturm and K1 for rings of real continuous functions

Abstract The paper contains a solution in the negative of the following problem of C.U. Jensen: Is the matrix cos t sin t -sin t cos t the product of 2×2 elementary matrices whose entries are continuous functions R → R ? The solution is based on some ideas of a proof of an of an old theorem of Sturm.