Generating the tame and wild kernels by Dennis-Stein symbols
Let \(F\) be a number field and \(S\) a finite set of primes of \(F\) containing the infinite primes. Let \(p\) be a prime number, \(e\) a positive integer, and \(n=p^ e\). Also, let \(L=F(\zeta_ n)\) and let \(T\) be the set of primes of \(L\) lying over \(S\). Let \({\mathcal O}_ S\subset F\) denote the \(S\)-integers of \(F\), i.e. \({\mathcal O}_ S=\{\alpha\in F\mid\;v_ \wp(\alpha)\geq 0\) for all \(\wp\not\in S\}\). Similarly for \({\mathcal O}_ T\subset L\). One has a homomorphism \[ J: \mu_ n\otimes\text{Cl}({\mathcal O}_ T)@>I>> K_ 2({\mathcal O}_ T)/n@>\text{tr}>> K_ 2({\mathcal O}_ S)/n, \] where tr denotes the transfer map. Write \({\mathcal H}\) for the maximal unramified abelian extension of \(L\) such that the finite primes of \(T\) split completely in \({\mathcal H}\). Then the main theorem of the paper says: Theorem. Let \(\varepsilon\in{\mathcal O}_ S^*\) and let \(k\) be such that \(\zeta_{p^ k}\in L\) but \(\zeta_{p^{k+1}}\not\in L\). If \(\root p \of {\varepsilon}\not\in L(\zeta_{p^{k+1}})\) and \(\zeta_{p^{k+1}}\not\in{\mathcal H}\), then the image of \(J\) is generated by classes of Dennis-Stein symbols. Several applications of the theorem are given, e.g. (*): for real quadratic number fields \(F=\mathbb{Q}(\sqrt d)\), \(d>1\) a square free integer such that \(d\not\equiv\pm 1 \pmod 8\), it follows that the 2-primary part of \(K_ 2({\mathcal O}_ F)\) is generated by Dennis-Stein symbols. In case \(\zeta_ p\not\in F\), \(e=1\), let \(N:{\mathcal O}_ T\to {\mathcal O}_ S\) be the norm map on ideal class groups. Also, let \(\Gamma=\text{Gal}(L/F)\). Consider the homomorphism \[ J':\mu_ p\otimes\text{Ker} (N) \to (\mu_ p\otimes\text{Cl}({\mathcal O}_ T))_ \Gamma@>I>> (K_ 2({\mathcal O}_ T)/p)_ \Gamma@>\text{tr}>> K_ 2({\mathcal O}_ S)/p. \] The theorem can no be improved to give: Theorem. Let \(\varepsilon\in{\mathcal O}_ S^*\). If \(\root p \of {\varepsilon}\not\in L\),then the image of \(J'\) is generated by classes of Dennis-Stein symbols. As a corollary one finds that if \(\#(S)>1\), then the image of \(J'\) is generated by classes of Dennis-Stein symbols. Again, several applications can be derived, e.g. (**): if \(p>3\) or \(d\not\equiv 6\pmod 9\), then the \(p\)-primary part of \(K_ 2({\mathcal O}_{\mathbb{Q}(\sqrt d)})\) is generated by Dennis-Stein symbols. Combining (*) and (**), one finds that, for a square free integer \(d\) such that \(d\not\equiv\pm 1\pmod 8\), and \(d\not\equiv 6\pmod 9\), \(K_ 2({\mathcal O}_{\mathbb{Q}(\sqrt d)})\) is generated by Dennis-Stein symbols. A list of fundamental units \(\varepsilon\) and generators, i.e. Dennis-Stein symbols, in terms of \(\varepsilon\), for \(K_ 2=(\mathbb{Q}(\sqrt d))\) for \(d\)=2, 3, 5, 7, 13, 21, 29, 69, 77, 82, 85, 93, 109, 178, 221, 229, 287, 406 is given. The final section states some results on the wild kernel \(W(F)\) of a number field \(F\), e.g. if \(F\) contains \(i\) or \(\sqrt{-2}\), then \(W(F)\) is generated by Dennis-Stein symbols.
scientific article; zbMATH DE number 89975