A generalization of the norm residue symbol. (English) Zbl 0576.12016
The norm residue symbol of degree n is a homomorphism \(\{\), \(\}\) : \(k^* \otimes_{{\mathbb{Z}}} k^*\to Br(k)_ n \otimes_{{\mathbb{Z}}} \mu_ n\) which for k containing the group \(\mu_ n\) of n th roots of unity is defined by \(\{a,b\}=(cl(A),\zeta)\) where \(\zeta\) is in \(\mu_ n\) and A is the central simple k-algebra which is the crossed product F[u], where \(F=k[z]\), \(z^ n=a\), is a Kummer extension of k, and u satisfies \(uz=\zeta zu\) and \(u^ n=b\), b defining a cyclic factor set for the cyclic group of order n.
This paper develops in explicit detail the idea that one can generalize the norm residue map to fields k not containing enough roots of unity by replacing the first factor of \(k^*\) (or \(k^*/k^{*n}\), on which \(\{\), \(\}\) is defined) by the group of J-Galois extensions of k, \(J={\mathbb{Z}}/n{\mathbb{Z}}\), which in turn has been identified by the author and M. E. Swedler [Hopf algebras and Galois theory (Lect. Notes Math. 97) (1969; Zbl 0197.014)] as \(Ext^ 1_{{\mathbb{Z}}\Delta}(\hat J, L^*)\) where \(L=k(\mu_ n)\), \(\Delta =Gal(L/k)\) and \(\hat J= the\) character group of J. \(\hat J\) can be replaced by \(({\mathbb{Z}}/n{\mathbb{Z}})^ t\), the \(\Delta\)-module \({\mathbb{Z}}/n{\mathbb{Z}}\) with action given by \(t: \Delta\) \(\to ({\mathbb{Z}}/n{\mathbb{Z}})^*\) where \(\delta (\zeta)=\zeta^{t(\delta)}\). This identification is a generalization of Kummer theory as found in, for example, N. Jacobson [Lectures in abstract algebra. III: Theory of fields and Galois theory (1980; Zbl 0455.12001)], the extension being obtained, in essence, via Galois descent from L to k.
Section 2 of the paper is devoted to a review of generalized Kummer theory of fields with Galois group J, an arbitrary finite abelian group. Section 3 examines the special case \(J={\mathbb{Z}}/n{\mathbb{Z}}\), and in describing extensions corresponding to J-Galois extensions of k uncovers some relations analogous to the classical Stickelberger relations on class groups of cyclotomic fields. The generalized norm residue map is defined in Section 4, and is shown to specialize properly to the known norm residue map when \(\mu_ n\) is in k, or to the norm residue map of local class field theory when k is a local field.
This paper develops in explicit detail the idea that one can generalize the norm residue map to fields k not containing enough roots of unity by replacing the first factor of \(k^*\) (or \(k^*/k^{*n}\), on which \(\{\), \(\}\) is defined) by the group of J-Galois extensions of k, \(J={\mathbb{Z}}/n{\mathbb{Z}}\), which in turn has been identified by the author and M. E. Swedler [Hopf algebras and Galois theory (Lect. Notes Math. 97) (1969; Zbl 0197.014)] as \(Ext^ 1_{{\mathbb{Z}}\Delta}(\hat J, L^*)\) where \(L=k(\mu_ n)\), \(\Delta =Gal(L/k)\) and \(\hat J= the\) character group of J. \(\hat J\) can be replaced by \(({\mathbb{Z}}/n{\mathbb{Z}})^ t\), the \(\Delta\)-module \({\mathbb{Z}}/n{\mathbb{Z}}\) with action given by \(t: \Delta\) \(\to ({\mathbb{Z}}/n{\mathbb{Z}})^*\) where \(\delta (\zeta)=\zeta^{t(\delta)}\). This identification is a generalization of Kummer theory as found in, for example, N. Jacobson [Lectures in abstract algebra. III: Theory of fields and Galois theory (1980; Zbl 0455.12001)], the extension being obtained, in essence, via Galois descent from L to k.
Section 2 of the paper is devoted to a review of generalized Kummer theory of fields with Galois group J, an arbitrary finite abelian group. Section 3 examines the special case \(J={\mathbb{Z}}/n{\mathbb{Z}}\), and in describing extensions corresponding to J-Galois extensions of k uncovers some relations analogous to the classical Stickelberger relations on class groups of cyclotomic fields. The generalized norm residue map is defined in Section 4, and is shown to specialize properly to the known norm residue map when \(\mu_ n\) is in k, or to the norm residue map of local class field theory when k is a local field.
Reviewer: L.N.Childs
MSC:
| 11S31 | Class field theory; \(p\)-adic formal groups |
| 14F22 | Brauer groups of schemes |
| 11S25 | Galois cohomology |
| 13B05 | Galois theory and commutative ring extensions |
| 12G05 | Galois cohomology |
Keywords:
norm residue symbol; roots of unity; central simple k-algebra; crossed product; Kummer extension; cyclic factor set; J-Galois extensions; character group; generalization of Kummer theory; Galois descent; Stickelberger relations; generalized norm residue map; local class field theoryReferences:
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